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mathematician, composer, photographer, fiddler

23 Apr 2021 | categories: Photographs, Mathematics, Film

Central Limit Theorem Music Video

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23 May 2020 | categories: Mathematics

Computing p-Typical Formal Group Laws

As part of my ongoing project with Doug Ravenel and Vitaly Lorman to compute the homotopy type of the string bordism spectrum $\mathrm{MO}\langle8\rangle$ at the prime 3, I wrote the following Mathematica code to efficiently compute the coefficients of p-typical formal group laws. It works at any prime p, with Araki or Hazewinkel generators. It also efficiently supports truncation, i.e. $\mathrm{BP}\langle k\rangle$. [This is a sliver of a larger library I have written for computing in Hopf rings.]

The code uses two methods. Method I is naïve but fast at low precision. Method II is more sophisticated, based on a perhaps not widely known theorem in Doug’s Green Book. It is generally slower at low precision but faster at high precision. My purpose here is both to make code for computing these things readily available, and to popularize this second method.

A little background: Formal group laws (henceforth FGL’s) are the sort of formal power series you get when you take the Taylor expansion of a Lie group at the identity. They lie between Lie groups and Lie algebras. There is a universal 1-dimensional FGL, and when you localize it at a prime p, it breaks up into simpler, p-typical FGL’s. These are FGL’s $F(x,y)$, also written $\newcommand{\Fplus}{+_F} x\Fplus y$, whose logarithms $\log_F(x)=\int_0^x\frac{dt}{d/dy(F(0,y))}$ have the form $\log_F(x)=\sum \ell_n x^{p^n}$. The sparsity of their coefficients makes them easier to work with.

Such FGL’s play an important role in algebraic topology, specifically chromatic homotopy theory. They describe how Chern classes within a given cohomology theory behave with respect to tensor product. And just as the universal 1-dimensional FGL splits up, when localized at a prime $p$, into $p$-typical formal group laws, the cohomology theory MU (complex cobordism) splits up, when localized, into simpler cohomology theories called BP (Brown-Peterson cohomology). The p-typical FGL’s computed below thus play an important role in computations involving BP and, via the Adams-Novikov spectral sequence, the homotopy groups of spheres.

There are two systems of generators commonly used when working with p-typical FGL’s – Araki and Hazewinkel, both denoted $v_i$, and defined by the nearly identical recursive formulas $ p\ell_n = \sum_{i=0}^n \ell_i v_{n-i}^{p^i}$ and $ p\ell_n = \sum_{i=0}^{n-1} \ell_i v_{n-i}^{p^i}$ respectively. The code supports both. Switch between them anytime by setting gen=Araki or gen=Hazewinkel. Similarly, switch between primes anytime by setting p=3 etc.

Back to the algorithms. Method I computes the p-typical FGL by the formula $$ x \Fplus y = \exp_F( \log_F(x)+\log_F(y) ). $$ This reduces the computation to off-the-shelf algorithms for the inverse and composition of power series.

Method II is based on Theorem 4.3.9 of the Green Book, which establishes an equality of p-typical formal sums, $$ {\sum_t}^F x_t = {\sum_I}^F v_I w_I(x_1,x_2,\dots). $$ On the RHS, the sum ranges over all finite sequences $I$ of positive integers, $v_I$ is a certain polynomial in the $v_i$’s of degree $2(p^{\|I\|}-1)$, and $w_I$ is a polynomial of degree $p^{\|I\|}$, where $\|I\|$ is the sum of the entries of $I$. (See Section 4.3 of the Green Book for details.)

This might not look like a promising way to compute $ x \Fplus y $, since it transforms it into an infinite formal sum of complicated polynomials, $$ x \Fplus y = v_{\emptyset}w_{\emptyset}(x,y) \Fplus v_{(1)}w_{(1)}(x,y) \Fplus v_{(2)}w_{(2)}(x,y) \Fplus v_{(1,1)}w_{(1,1)}(x,y) \Fplus \cdots $$ But it has its pluses! If you are only working to some fixed precision, say, then since the degrees of the polynomials $w_I$ grow quickly, only a few of them matter. In fact, you can apply the theorem a second time to obtain $$ x \Fplus y = v_{\emptyset}w_{\emptyset}(v_{\emptyset}w_{\emptyset}(x,y),w_{(1)}(x,y),v_{(2)}w_{(2)}(x,y),\dots) \Fplus v_{(1)}w_{(1)}(\dots) \Fplus v_{(2)}w_{(2)}(\dots)\Fplus \cdots$$ Again, this might not look promising. In fact, it might even look crazy. But now the degrees of the formal summands (or rather their vanishing orders—they are no longer homogeneous) grow even more quickly than before. By applying the theorem again and again, you can make each summand except the first have such high degree that it is irrelevant to your working precision, and thereby eliminate formal sums from the RHS altogether. That’s exactly how the code below works. [The code is also available from github.]

p=2; gen=Araki; gen=Hazewinkel;

BPl[0,_.,_.,_.] = 1;
BPl[n_Integer,k:(_Integer|Infinity):Infinity] := BPl[n,k,p,gen];
BPl[n_Integer,k:(_Integer|Infinity),p_Integer,gen_Symbol] := BPl[n,k,p,gen] =
  Simplify[1/(p-If[gen===Araki,p^p^n,0])
Sum[BPl[i,k,p,gen] Subscript[v,n-i]^p^i, {i,Max[n-k,0],n-1}] /. Subscript[v,0]->p ];
logBP[ord_Integer,k:(_Integer|Infinity):Infinity] := logBP[ord,k,p,gen];
logBP[ord_Integer,k_,p_Integer,gen_Symbol] := logBP[ord,k,p,gen] =
  Function[d,Evaluate[Sum[BPl[n,k,p,gen] d^p^n, {n, 0, Log[p, ord]}] + O[d]^ord]];
expBP[ord_Integer,k:(_Integer|Infinity):Infinity] := expBP[ord,k,p,gen];
expBP[ord_Integer,k_,p_Integer,gen_Symbol] := expBP[ord,k,p,gen] =
  Function[d,Evaluate[Map[Simplify,InverseSeries[logBP[ord,k,p,gen][d]],{2}]]];
fglBP[ord_Integer,k:(_Integer|Infinity):Infinity,d_Symbol:d] := fglBP[ord,k,d,p,gen];
fglBP[ord_Integer,k_,d_Symbol,p_Integer,gen_Symbol] :=
  fglBP[ord,k,d,p,gen]=Function[{x,y},Evaluate[Module[{z}, Map[Expand, 
     ComposeSeries[expBP[ord,k,p,gen][d],
       ComposeSeries[logBP[ord,k,p,gen][z],x d+O[d]^ord] +
       ComposeSeries[logBP[ord,k,p,gen][z],y d+O[d]^ord]], {2}]]]];
pSerBP[ord_Integer,k:(_Integer|Infinity):Infinity,d_Symbol:d]:=pSerBP[ord,k,d,p,gen];
pSerBP[ord_Integer,k_,d_Symbol,p_Integer,gen_Symbol][x_] :=
  Module[{z}, With[{xPlus=(Normal[fglBP[ord,k,d,p,gen][x,z]]/.d->1)+O[z]^ord},
    Map[Expand,Nest[ComposeSeries[xPlus,#]&,x+O[x]^ord,p-1],{2}]]];

vI[I_List] := vI[I,p]; vI[{},_] = 1;
vI[I_List,p_Integer] := vI[I,p] =
  Subscript[v, First[I]] vI[Rest[I]]^p^First[I] /. Subscript[v,0]->p;
II[I_List] := II[I,p]; II[n_Integer,p_Integer] := II[n,p]=p-p^p^n;
II[{},_]=1; II[I_List,p_Integer] := II[I,p]=II[Plus@@I,p]II[Most[I],p];

w[K_List,nvars_Integer] := w[K,nvars,p,gen];
w[{},_,_,_][vars__] := Plus[vars];
w[K_List,nvars_Integer,p_Integer,gen_Symbol] :=
  With[{slotvars = Slot/@Range[1,nvars]},
  With[{formula  = 1/If[gen===Hazewinkel, p^Length[K], II[K]]
    (Plus@@(#^p^(Plus@@K) & /@ slotvars)
     - Plus@@(Function[j,With[{I=Drop[K,-j],J=Take[K,-j]},
               If[gen===Hazewinkel, p^Length[J], II[K]/II[I]]
               w[J,nvars,p,gen][Sequence@@slotvars]^p^(Plus@@I)]]
                 /@ Range[0,Length[K]-1]))},
      w[K,nvars,p,gen]=formula&; formula& ]];

BPSumSimplify[ord_Integer,k:(_Integer|Infinity):Infinity,d_Symbol:d] :=
  BPSumSimplify[ord,k,d,p,gen];
BPSumSimplify[ord_Integer,k:(_Integer|Infinity),
              d_Symbol,p_Integer,gen_Symbol][terms_List] :=
  With[{mintermvord = Min[Exponent[#,d,Min]& /@ terms]},
    Cases[
     Map[Simplify, vI[#] d^(2(p^(Plus@@#)-1))
                   (w[#,Length[terms],p,gen]@@terms) + O[d]^ord, {2}]& /@
           Flatten[Permutations/@IntegerPartitions[#,All,Range[1,Min[k,#]]]&/@
                                     Range[0,Log[p,(1+ord)/(2+mintermvord)]],2],
     Except[O[d]^ord]]];

BPSumToOrder[ord_Integer,k:(_Integer|Infinity):Infinity,d_Symbol:d] :=
   BPSumToOrder[ord,k,d,p,gen];
BPSumToOrder[ord_Integer,k:(_Integer|Infinity),
     d_Symbol,p_Integer,gen_Symbol][terms__] :=
  Map[Expand,First[NestWhile[BPSumSimplify[ord,k,d,p,gen], {terms},
                             Length[#]>1&],{2}]];

pSeriesBP[ord_Integer,k:(_Integer|Infinity):Infinity,d_Symbol:d] :=
  pSeriesBP[ord,k,d,p,gen];
pSeriesBP[ord_Integer,k:(_Integer|Infinity),
          d_Symbol,p_Integer,gen_Symbol][x_] :=
 BPSumToOrder[ord,k,d,p,gen]@@Table[x, p];

To warm up, let’s compute the 2-typical FGL in terms of Hazewinkel generators to degree 12 by running

p=2; gen=Hazewinkel;
fglBP[13][x,y]

which outputs

$d (x+y)-d^2 v_1 x y+d^3 \left(v_1^2 x^2 y+v_1^2 x y^2\right)+d^4 \left(\left(-2 v_1^3-2 v_2\right) x^3 y+\left(-4 v_1^3-3 v_2\right) x^2 y^2+\left(-2 v_1^3-2 v_2\right) x y^3\right)+d^5 \left(\left(3 v_1^4+4 v_2 v_1\right) x^4 y+\left(10 v_1^4+11 v_2 v_1\right) x^3 y^2+\left(10 v_1^4+11 v_2 v_1\right) x^2 y^3+\left(3 v_1^4+4 v_2 v_1\right) x y^4\right)+d^6 \left(\left(-4 v_1^5-6 v_2 v_1^2\right) x^5 y+\left(-21 v_1^5-28 v_2 v_1^2\right) x^4 y^2+\left(-34 v_1^5-43 v_2 v_1^2\right) x^3 y^3+\left(-21 v_1^5-28 v_2 v_1^2\right) x^2 y^4+\left(-4 v_1^5-6 v_2 v_1^2\right) x y^5\right)+d^7 \left(\left(6 v_1^6+12 v_2 v_1^3+4 v_2^2\right) x^6 y+\left(43 v_1^6+75 v_2 v_1^3+18 v_2^2\right) x^5 y^2+\left(101 v_1^6+164 v_2 v_1^3+34 v_2^2\right) x^4 y^3+\left(101 v_1^6+164 v_2 v_1^3+34 v_2^2\right) x^3 y^4+\left(43 v_1^6+75 v_2 v_1^3+18 v_2^2\right) x^2 y^5+\left(6 v_1^6+12 v_2 v_1^3+4 v_2^2\right) x y^6\right)+d^8 \left(\left(-10 v_1^7-24 v_2 v_1^4-14 v_2^2 v_1-4 v_3\right) x^7 y+\left(-88 v_1^7-190 v_2 v_1^4-89 v_2^2 v_1-14 v_3\right) x^6 y^2+\left(-275 v_1^7-551 v_2 v_1^4-226 v_2^2 v_1-28 v_3\right) x^5 y^3+\left(-394 v_1^7-769 v_2 v_1^4-302 v_2^2 v_1-35 v_3\right) x^4 y^4+\left(-275 v_1^7-551 v_2 v_1^4-226 v_2^2 v_1-28 v_3\right) x^3 y^5+\left(-88 v_1^7-190 v_2 v_1^4-89 v_2^2 v_1-14 v_3\right) x^2 y^6+\left(-10 v_1^7-24 v_2 v_1^4-14 v_2^2 v_1-4 v_3\right) x y^7\right)+d^9 \left(\left(15 v_1^8+40 v_2 v_1^5+28 v_2^2 v_1^2+8 v_3 v_1\right) x^8 y+\left(169 v_1^8+420 v_2 v_1^5+257 v_2^2 v_1^2+46 v_3 v_1\right) x^7 y^2+\left(680 v_1^8+1586 v_2 v_1^5+879 v_2^2 v_1^2+126 v_3 v_1\right) x^6 y^3+\left(1303 v_1^8+2933 v_2 v_1^5+1543 v_2^2 v_1^2+203 v_3 v_1\right) x^5 y^4+\left(1303 v_1^8+2933 v_2 v_1^5+1543 v_2^2 v_1^2+203 v_3 v_1\right) x^4 y^5+\left(680 v_1^8+1586 v_2 v_1^5+879 v_2^2 v_1^2+126 v_3 v_1\right) x^3 y^6+\left(169 v_1^8+420 v_2 v_1^5+257 v_2^2 v_1^2+46 v_3 v_1\right) x^2 y^7+\left(15 v_1^8+40 v_2 v_1^5+28 v_2^2 v_1^2+8 v_3 v_1\right) x y^8\right)+d^{10} \left(\left(-22 v_1^9-66 v_2 v_1^6-58 v_2^2 v_1^3-12 v_3 v_1^2-8 v_2^3\right) x^9 y+\left(-312 v_1^9-880 v_2 v_1^6-688 v_2^2 v_1^3-104 v_3 v_1^2-72 v_2^3\right) x^8 y^2+\left(-1573 v_1^9-4192 v_2 v_1^6-3001 v_2^2 v_1^3-382 v_3 v_1^2-260 v_2^3\right) x^7 y^3+\left(-3861 v_1^9-9900 v_2 v_1^6-6707 v_2^2 v_1^3-791 v_3 v_1^2-523 v_2^3\right) x^6 y^4+\left(-5156 v_1^9-13042 v_2 v_1^6-8671 v_2^2 v_1^3-1001 v_3 v_1^2-654 v_2^3\right) x^5 y^5+\left(-3861 v_1^9-9900 v_2 v_1^6-6707 v_2^2 v_1^3-791 v_3 v_1^2-523 v_2^3\right) x^4 y^6+\left(-1573 v_1^9-4192 v_2 v_1^6-3001 v_2^2 v_1^3-382 v_3 v_1^2-260 v_2^3\right) x^3 y^7+\left(-312 v_1^9-880 v_2 v_1^6-688 v_2^2 v_1^3-104 v_3 v_1^2-72 v_2^3\right) x^2 y^8+\left(-22 v_1^9-66 v_2 v_1^6-58 v_2^2 v_1^3-12 v_3 v_1^2-8 v_2^3\right) x y^9\right)+d^{11} \left(\left(34 v_1^{10}+116 v_2 v_1^7+128 v_2^2 v_1^4+24 v_3 v_1^3+40 v_2^3 v_1+16 v_2 v_3\right) x^{10} y+\left(574 v_1^{10}+1837 v_2 v_1^7+1811 v_2^2 v_1^4+254 v_3 v_1^3+456 v_2^3 v_1+120 v_2 v_3\right) x^9 y^2+\left(3506 v_1^{10}+10612 v_2 v_1^7+9596 v_2^2 v_1^4+1144 v_3 v_1^3+2060 v_2^3 v_1+424 v_2 v_3\right) x^8 y^3+\left(10643 v_1^{10}+30921 v_2 v_1^7+26363 v_2^2 v_1^4+2906 v_3 v_1^3+5103 v_2^3 v_1+918 v_2 v_3\right) x^7 y^4+\left(18115 v_1^{10}+51502 v_2 v_1^7+42619 v_2^2 v_1^4+4557 v_3 v_1^3+7847 v_2^3 v_1+1330 v_2 v_3\right) x^6 y^5+\left(18115 v_1^{10}+51502 v_2 v_1^7+42619 v_2^2 v_1^4+4557 v_3 v_1^3+7847 v_2^3 v_1+1330 v_2 v_3\right) x^5 y^6+\left(10643 v_1^{10}+30921 v_2 v_1^7+26363 v_2^2 v_1^4+2906 v_3 v_1^3+5103 v_2^3 v_1+918 v_2 v_3\right) x^4 y^7+\left(3506 v_1^{10}+10612 v_2 v_1^7+9596 v_2^2 v_1^4+1144 v_3 v_1^3+2060 v_2^3 v_1+424 v_2 v_3\right) x^3 y^8+\left(574 v_1^{10}+1837 v_2 v_1^7+1811 v_2^2 v_1^4+254 v_3 v_1^3+456 v_2^3 v_1+120 v_2 v_3\right) x^2 y^9+\left(34 v_1^{10}+116 v_2 v_1^7+128 v_2^2 v_1^4+24 v_3 v_1^3+40 v_2^3 v_1+16 v_2 v_3\right) x y^{10}\right)+d^{12} \left(\left(-52 v_1^{11}-196 v_2 v_1^8-250 v_2^2 v_1^5-44 v_3 v_1^4-104 v_2^3 v_1^2-48 v_2 v_3 v_1\right) x^{11} y+\left(-1039 v_1^{11}-3704 v_2 v_1^8-4325 v_2^2 v_1^5-582 v_3 v_1^4-1564 v_2^3 v_1^2-520 v_2 v_3 v_1\right) x^{10} y^2+\left(-7546 v_1^{11}-25549 v_2 v_1^8-27679 v_2^2 v_1^5-3170 v_3 v_1^4-8920 v_2^3 v_1^2-2400 v_2 v_3 v_1\right) x^9 y^3+\left(-27636 v_1^{11}-89822 v_2 v_1^8-91989 v_2^2 v_1^5-9682 v_3 v_1^4-27241 v_2^3 v_1^2-6470 v_2 v_3 v_1\right) x^8 y^4+\left(-58002 v_1^{11}-183665 v_2 v_1^8-181684 v_2^2 v_1^5-18408 v_3 v_1^4-51133 v_2^3 v_1^2-11384 v_2 v_3 v_1\right) x^7 y^5+\left(-73850 v_1^{11}-231765 v_2 v_1^8-226598 v_2^2 v_1^5-22715 v_3 v_1^4-62707 v_2^3 v_1^2-13685 v_2 v_3 v_1\right) x^6 y^6+\left(-58002 v_1^{11}-183665 v_2 v_1^8-181684 v_2^2 v_1^5-18408 v_3 v_1^4-51133 v_2^3 v_1^2-11384 v_2 v_3 v_1\right) x^5 y^7+\left(-27636 v_1^{11}-89822 v_2 v_1^8-91989 v_2^2 v_1^5-9682 v_3 v_1^4-27241 v_2^3 v_1^2-6470 v_2 v_3 v_1\right) x^4 y^8+\left(-7546 v_1^{11}-25549 v_2 v_1^8-27679 v_2^2 v_1^5-3170 v_3 v_1^4-8920 v_2^3 v_1^2-2400 v_2 v_3 v_1\right) x^3 y^9+\left(-1039 v_1^{11}-3704 v_2 v_1^8-4325 v_2^2 v_1^5-582 v_3 v_1^4-1564 v_2^3 v_1^2-520 v_2 v_3 v_1\right) x^2 y^{10}+\left(-52 v_1^{11}-196 v_2 v_1^8-250 v_2^2 v_1^5-44 v_3 v_1^4-104 v_2^3 v_1^2-48 v_2 v_3 v_1\right) x y^{11}\right)+O\left(d^{13}\right)$

[Mathematica handles single-variable power series better than multi-variable ones, so the code represents this two-variable power series in $x$ and $y$ as a power series in a single dummy variable $d$.]

fglBP and the rest of the commands take an optional second parameter k (which defaults to ∞) for restricting to $\mathrm{BP}\langle k\rangle$. Put simply, this kills $v_n$ for $n>k$. Let’s go to $\mathrm{BP}\langle1\rangle$ and increase precision, remaining at p=2,

fglBP[24,1][x,y]

to get

$d (x+y)-d^2 v_1 x y+d^3 \left(v_1^2 x^2 y+v_1^2 x y^2\right)+d^4 \left(-2 v_1^3 x^3 y-4 v_1^3 x^2 y^2-2 v_1^3 x y^3\right)+d^5 \left(3 v_1^4 x^4 y+10 v_1^4 x^3 y^2+10 v_1^4 x^2 y^3+3 v_1^4 x y^4\right)+d^6 \left(-4 v_1^5 x^5 y-21 v_1^5 x^4 y^2-34 v_1^5 x^3 y^3-21 v_1^5 x^2 y^4-4 v_1^5 x y^5\right)+d^7 \left(6 v_1^6 x^6 y+43 v_1^6 x^5 y^2+101 v_1^6 x^4 y^3+101 v_1^6 x^3 y^4+43 v_1^6 x^2 y^5+6 v_1^6 x y^6\right)+d^8 \left(-10 v_1^7 x^7 y-88 v_1^7 x^6 y^2-275 v_1^7 x^5 y^3-394 v_1^7 x^4 y^4-275 v_1^7 x^3 y^5-88 v_1^7 x^2 y^6-10 v_1^7 x y^7\right)+d^9 \left(15 v_1^8 x^8 y+169 v_1^8 x^7 y^2+680 v_1^8 x^6 y^3+1303 v_1^8 x^5 y^4+1303 v_1^8 x^4 y^5+680 v_1^8 x^3 y^6+169 v_1^8 x^2 y^7+15 v_1^8 x y^8\right)+d^{10} \left(-22 v_1^9 x^9 y-312 v_1^9 x^8 y^2-1573 v_1^9 x^7 y^3-3861 v_1^9 x^6 y^4-5156 v_1^9 x^5 y^5-3861 v_1^9 x^4 y^6-1573 v_1^9 x^3 y^7-312 v_1^9 x^2 y^8-22 v_1^9 x y^9\right)+d^{11} \left(34 v_1^{10} x^{10} y+574 v_1^{10} x^9 y^2+3506 v_1^{10} x^8 y^3+10643 v_1^{10} x^7 y^4+18115 v_1^{10} x^6 y^5+18115 v_1^{10} x^5 y^6+10643 v_1^{10} x^4 y^7+3506 v_1^{10} x^3 y^8+574 v_1^{10} x^2 y^9+34 v_1^{10} x y^{10}\right)+d^{12} \left(-52 v_1^{11} x^{11} y-1039 v_1^{11} x^{10} y^2-7546 v_1^{11} x^9 y^3-27636 v_1^{11} x^8 y^4-58002 v_1^{11} x^7 y^5-73850 v_1^{11} x^6 y^6-58002 v_1^{11} x^5 y^7-27636 v_1^{11} x^4 y^8-7546 v_1^{11} x^3 y^9-1039 v_1^{11} x^2 y^{10}-52 v_1^{11} x y^{11}\right)+d^{13} \left(78 v_1^{12} x^{12} y+1840 v_1^{12} x^{11} y^2+15709 v_1^{12} x^{10} y^3+68193 v_1^{12} x^9 y^4+172287 v_1^{12} x^8 y^5+270163 v_1^{12} x^7 y^6+270163 v_1^{12} x^6 y^7+172287 v_1^{12} x^5 y^8+68193 v_1^{12} x^4 y^9+15709 v_1^{12} x^3 y^{10}+1840 v_1^{12} x^2 y^{11}+78 v_1^{12} x y^{12}\right)+d^{14} \left(-118 v_1^{13} x^{13} y-3224 v_1^{13} x^{12} y^2-31900 v_1^{13} x^{11} y^3-161602 v_1^{13} x^{10} y^4-482163 v_1^{13} x^9 y^5-908378 v_1^{13} x^8 y^6-1118046 v_1^{13} x^7 y^7-908378 v_1^{13} x^6 y^8-482163 v_1^{13} x^5 y^9-161602 v_1^{13} x^4 y^{10}-31900 v_1^{13} x^3 y^{11}-3224 v_1^{13} x^2 y^{12}-118 v_1^{13} x y^{13}\right)+d^{15} \left(180 v_1^{14} x^{14} y+5611 v_1^{14} x^{13} y^2+63524 v_1^{14} x^{12} y^3+370532 v_1^{14} x^{11} y^4+1285316 v_1^{14} x^{10} y^5+2853441 v_1^{14} x^9 y^6+4215142 v_1^{14} x^8 y^7+4215142 v_1^{14} x^7 y^8+2853441 v_1^{14} x^6 y^9+1285316 v_1^{14} x^5 y^{10}+370532 v_1^{14} x^4 y^{11}+63524 v_1^{14} x^3 y^{12}+5611 v_1^{14} x^2 y^{13}+180 v_1^{14} x y^{14}\right)+d^{16} \left(-274 v_1^{15} x^{15} y-9675 v_1^{15} x^{14} y^2-124223 v_1^{15} x^{13} y^3-825517 v_1^{15} x^{12} y^4-3287502 v_1^{15} x^{11} y^5-8468689 v_1^{15} x^{10} y^6-14724994 v_1^{15} x^9 y^7-17666224 v_1^{15} x^8 y^8-14724994 v_1^{15} x^7 y^9-8468689 v_1^{15} x^6 y^{10}-3287502 v_1^{15} x^5 y^{11}-825517 v_1^{15} x^4 y^{12}-124223 v_1^{15} x^3 y^{13}-9675 v_1^{15} x^2 y^{14}-274 v_1^{15} x y^{15}\right)+d^{17} \left(415 v_1^{16} x^{16} y+16531 v_1^{16} x^{15} y^2+238981 v_1^{16} x^{14} y^3+1793551 v_1^{16} x^{13} y^4+8114767 v_1^{16} x^{12} y^5+23952980 v_1^{16} x^{11} y^6+48268807 v_1^{16} x^{10} y^7+68123059 v_1^{16} x^9 y^8+68123059 v_1^{16} x^8 y^9+48268807 v_1^{16} x^7 y^{10}+23952980 v_1^{16} x^6 y^{11}+8114767 v_1^{16} x^5 y^{12}+1793551 v_1^{16} x^4 y^{13}+238981 v_1^{16} x^3 y^{14}+16531 v_1^{16} x^2 y^{15}+415 v_1^{16} x y^{16}\right)+d^{18} \left(-630 v_1^{17} x^{17} y-28072 v_1^{17} x^{16} y^2-453515 v_1^{17} x^{15} y^3-3813416 v_1^{17} x^{14} y^4-19426112 v_1^{17} x^{13} y^5-65016261 v_1^{17} x^{12} y^6-149931112 v_1^{17} x^{11} y^7-245052157 v_1^{17} x^{10} y^8-288197074 v_1^{17} x^9 y^9-245052157 v_1^{17} x^8 y^{10}-149931112 v_1^{17} x^7 y^{11}-65016261 v_1^{17} x^6 y^{12}-19426112 v_1^{17} x^5 y^{13}-3813416 v_1^{17} x^4 y^{14}-453515 v_1^{17} x^3 y^{15}-28072 v_1^{17} x^2 y^{16}-630 v_1^{17} x y^{17}\right)+d^{19} \left(958 v_1^{18} x^{18} y+47412 v_1^{18} x^{17} y^2+850384 v_1^{18} x^{16} y^3+7955817 v_1^{18} x^{15} y^4+45277700 v_1^{18} x^{14} y^5+170286221 v_1^{18} x^{13} y^6+444652182 v_1^{18} x^{12} y^7+830961623 v_1^{18} x^{11} y^8+1131282117 v_1^{18} x^{10} y^9+1131282117 v_1^{18} x^9 y^{10}+830961623 v_1^{18} x^8 y^{11}+444652182 v_1^{18} x^7 y^{12}+170286221 v_1^{18} x^6 y^{13}+45277700 v_1^{18} x^5 y^{14}+7955817 v_1^{18} x^4 y^{15}+850384 v_1^{18} x^3 y^{16}+47412 v_1^{18} x^2 y^{17}+958 v_1^{18} x y^{18}\right)+d^{20} \left(-1454 v_1^{19} x^{19} y-79606 v_1^{19} x^{18} y^2-1576972 v_1^{19} x^{17} y^3-16318194 v_1^{19} x^{16} y^4-103060418 v_1^{19} x^{15} y^5-432229539 v_1^{19} x^{14} y^6-1266616868 v_1^{19} x^{13} y^7-2677879750 v_1^{19} x^{12} y^8-4166290333 v_1^{19} x^{11} y^9-4822077912 v_1^{19} x^{10} y^{10}-4166290333 v_1^{19} x^9 y^{11}-2677879750 v_1^{19} x^8 y^{12}-1266616868 v_1^{19} x^7 y^{13}-432229539 v_1^{19} x^6 y^{14}-103060418 v_1^{19} x^5 y^{15}-16318194 v_1^{19} x^4 y^{16}-1576972 v_1^{19} x^3 y^{17}-79606 v_1^{19} x^2 y^{18}-1454 v_1^{19} x y^{19}\right)+d^{21} \left(2206 v_1^{20} x^{20} y+132983 v_1^{20} x^{19} y^2+2895438 v_1^{20} x^{18} y^3+32964477 v_1^{20} x^{17} y^4+229684072 v_1^{20} x^{16} y^5+1067054994 v_1^{20} x^{15} y^6+3482396871 v_1^{20} x^{14} y^7+8254959638 v_1^{20} x^{13} y^8+14521182350 v_1^{20} x^{12} y^9+19201231409 v_1^{20} x^{11} y^{10}+19201231409 v_1^{20} x^{10} y^{11}+14521182350 v_1^{20} x^9 y^{12}+8254959638 v_1^{20} x^8 y^{13}+3482396871 v_1^{20} x^7 y^{14}+1067054994 v_1^{20} x^6 y^{15}+229684072 v_1^{20} x^5 y^{16}+32964477 v_1^{20} x^4 y^{17}+2895438 v_1^{20} x^3 y^{18}+132983 v_1^{20} x^2 y^{19}+2206 v_1^{20} x y^{20}\right)+d^{22} \left(-3350 v_1^{21} x^{21} y-221224 v_1^{21} x^{20} y^2-5269567 v_1^{21} x^{19} y^3-65689701 v_1^{21} x^{18} y^4-502303029 v_1^{21} x^{17} y^5-2569852892 v_1^{21} x^{16} y^6-9278277763 v_1^{21} x^{15} y^7-24471261938 v_1^{21} x^{14} y^8-48234231310 v_1^{21} x^{13} y^9-72085472634 v_1^{21} x^{12} y^{10}-82344826740 v_1^{21} x^{11} y^{11}-72085472634 v_1^{21} x^{10} y^{12}-48234231310 v_1^{21} x^9 y^{13}-24471261938 v_1^{21} x^8 y^{14}-9278277763 v_1^{21} x^7 y^{15}-2569852892 v_1^{21} x^6 y^{16}-502303029 v_1^{21} x^5 y^{17}-65689701 v_1^{21} x^4 y^{18}-5269567 v_1^{21} x^3 y^{19}-221224 v_1^{21} x^2 y^{20}-3350 v_1^{21} x y^{21}\right)+d^{23} \left(5088 v_1^{22} x^{22} y+366561 v_1^{22} x^{21} y^2+9513514 v_1^{22} x^{20} y^3+129295256 v_1^{22} x^{19} y^4+1079943317 v_1^{22} x^{18} y^5+6053073238 v_1^{22} x^{17} y^6+24036231782 v_1^{22} x^{16} y^7+70066929304 v_1^{22} x^{15} y^8+153557898052 v_1^{22} x^{14} y^9+257032174895 v_1^{22} x^{13} y^{10}+331778552977 v_1^{22} x^{12} y^{11}+331778552977 v_1^{22} x^{11} y^{12}+257032174895 v_1^{22} x^{10} y^{13}+153557898052 v_1^{22} x^9 y^{14}+70066929304 v_1^{22} x^8 y^{15}+24036231782 v_1^{22} x^7 y^{16}+6053073238 v_1^{22} x^6 y^{17}+1079943317 v_1^{22} x^5 y^{18}+129295256 v_1^{22} x^4 y^{19}+9513514 v_1^{22} x^3 y^{20}+366561 v_1^{22} x^2 y^{21}+5088 v_1^{22} x y^{22}\right)+O\left(d^{24}\right)$

So far we have only used Method I. To perform the same computation using Method II, we would run BPSumToOrder[45,1][x,y]. (We need to compute to order 45 to obtain the same result, because, while fglBP places $x$ and $y$ in degree $1$ and each $v_i$ in degree $0$, BPSumToOrder places $x$ and $y$ in degree $0$ and each $v_i$ in degree $2(p^i-1)$. You can place $x$ or $y$ in any degree $n$ by replacing it with the monomial $xd^n$.)

Method II excels at high-precision calculations, and at handling more than two summands. Simply list the additional summands as additional parameters to BPSumToOrder, e.g. to compute $x +_F y +_F z$ to topological order 30, call BPSumToOrder[30][x,y,z]. For example, formal sums like $$1 +_F t_1 +_F t_2 +_F t_3,$$ where $t_i$ has topological degree $2(p^i-1)$, can be efficiently computed to topological order 30, remaining at p=2, by running

BPSumToOrder[30][1, Subscript[t,1] d^2, Subscript[t,2] d^6, Subscript[t,3] d^14]

which outputs

$1+d^2 t_1-d^4 t_1 v_1+d^6 \left(t_1 v_1^2+t_2\right)+d^8 \left(-2 t_1 v_1^3+t_1^2 v_1^2-t_2 v_1-2 t_1 v_2\right)+d^{10} \left(3 t_1 v_1^4-4 t_1^2 v_1^3+t_2 v_1^2-t_1 t_2 v_1+4 t_1 v_2 v_1-3 t_1^2 v_2\right)+d^{12} \left(-4 t_1 v_1^5+10 t_1^2 v_1^4-2 t_1^3 v_1^3-2 t_2 v_1^3+3 t_1 t_2 v_1^2-6 t_1 v_2 v_1^2+11 t_1^2 v_2 v_1-2 t_1^3 v_2-2 t_2 v_2\right)+d^{14} \left(6 t_1 v_1^6-21 t_1^2 v_1^5+10 t_1^3 v_1^4+3 t_2 v_1^4-9 t_1 t_2 v_1^3+12 t_1 v_2 v_1^3+t_1^2 t_2 v_1^2-28 t_1^2 v_2 v_1^2+11 t_1^3 v_2 v_1+4 t_2 v_2 v_1+4 t_1 v_2^2-6 t_1 t_2 v_2+t_3\right)+d^{16} \left(-10 t_1 v_1^7+43 t_1^2 v_1^6-34 t_1^3 v_1^5-4 t_2 v_1^5+3 t_1^4 v_1^4+22 t_1 t_2 v_1^4-24 t_1 v_2 v_1^4-9 t_1^2 t_2 v_1^3+75 t_1^2 v_2 v_1^3+t_2^2 v_1^2-43 t_1^3 v_2 v_1^2-6 t_2 v_2 v_1^2-14 t_1 v_2^2 v_1-t_3 v_1+4 t_1^4 v_2 v_1+24 t_1 t_2 v_2 v_1+18 t_1^2 v_2^2-6 t_1^2 t_2 v_2-4 t_1 v_3\right)+d^{18} \left(15 t_1 v_1^8-88 t_1^2 v_1^7+101 t_1^3 v_1^6+6 t_2 v_1^6-21 t_1^4 v_1^5-45 t_1 t_2 v_1^5+40 t_1 v_2 v_1^5+39 t_1^2 t_2 v_1^4-190 t_1^2 v_2 v_1^4-4 t_2^2 v_1^3-2 t_1^3 t_2 v_1^3+164 t_1^3 v_2 v_1^3+12 t_2 v_2 v_1^3+t_1 t_2^2 v_1^2+28 t_1 v_2^2 v_1^2+t_3 v_1^2-28 t_1^4 v_2 v_1^2-60 t_1 t_2 v_2 v_1^2-89 t_1^2 v_2^2 v_1-t_1 t_3 v_1+39 t_1^2 t_2 v_2 v_1+8 t_1 v_3 v_1+34 t_1^3 v_2^2+4 t_2 v_2^2-3 t_2^2 v_2-2 t_1^3 t_2 v_2-14 t_1^2 v_3\right)+d^{20} \left(-22 t_1 v_1^9+169 t_1^2 v_1^8-275 t_1^3 v_1^7-10 t_2 v_1^7+101 t_1^4 v_1^6+90 t_1 t_2 v_1^6-66 t_1 v_2 v_1^6-4 t_1^5 v_1^5-124 t_1^2 t_2 v_1^5+420 t_1^2 v_2 v_1^5+10 t_2^2 v_1^4+22 t_1^3 t_2 v_1^4-551 t_1^3 v_2 v_1^4-24 t_2 v_2 v_1^4-9 t_1 t_2^2 v_1^3-58 t_1 v_2^2 v_1^3-2 t_3 v_1^3+164 t_1^4 v_2 v_1^3+156 t_1 t_2 v_2 v_1^3+257 t_1^2 v_2^2 v_1^2+3 t_1 t_3 v_1^2-6 t_1^5 v_2 v_1^2-153 t_1^2 t_2 v_2 v_1^2-12 t_1 v_3 v_1^2-226 t_1^3 v_2^2 v_1-14 t_2 v_2^2 v_1+11 t_2^2 v_2 v_1+24 t_1^3 t_2 v_2 v_1+46 t_1^2 v_3 v_1-8 t_1 v_2^3+34 t_1^4 v_2^2+36 t_1 t_2 v_2^2-6 t_1 t_2^2 v_2-2 t_3 v_2-28 t_1^3 v_3-4 t_2 v_3\right)+d^{22} \left(34 t_1 v_1^{10}-312 t_1^2 v_1^9+680 t_1^3 v_1^8+15 t_2 v_1^8-394 t_1^4 v_1^7-182 t_1 t_2 v_1^7+116 t_1 v_2 v_1^7+43 t_1^5 v_1^6+348 t_1^2 t_2 v_1^6-880 t_1^2 v_2 v_1^6-21 t_2^2 v_1^5-124 t_1^3 t_2 v_1^5+1586 t_1^3 v_2 v_1^5+40 t_2 v_2 v_1^5+39 t_1 t_2^2 v_1^4+128 t_1 v_2^2 v_1^4+3 t_1^4 t_2 v_1^4+3 t_3 v_1^4-769 t_1^4 v_2 v_1^4-392 t_1 t_2 v_2 v_1^4-4 t_1^2 t_2^2 v_1^3-688 t_1^2 v_2^2 v_1^3-9 t_1 t_3 v_1^3+75 t_1^5 v_2 v_1^3+552 t_1^2 t_2 v_2 v_1^3+24 t_1 v_3 v_1^3+879 t_1^3 v_2^2 v_1^2+28 t_2 v_2^2 v_1^2+t_1^2 t_3 v_1^2-28 t_2^2 v_2 v_1^2-153 t_1^3 t_2 v_2 v_1^2-104 t_1^2 v_3 v_1^2+40 t_1 v_2^3 v_1-302 t_1^4 v_2^2 v_1-182 t_1 t_2 v_2^2 v_1-t_2 t_3 v_1+39 t_1 t_2^2 v_2 v_1+4 t_1^4 t_2 v_2 v_1+4 t_3 v_2 v_1+126 t_1^3 v_3 v_1+8 t_2 v_3 v_1-72 t_1^2 v_2^3+18 t_1^5 v_2^2+102 t_1^2 t_2 v_2^2-3 t_1^2 t_2^2 v_2-6 t_1 t_3 v_2-35 t_1^4 v_3-28 t_1 t_2 v_3+16 t_1 v_2 v_3\right)+\left(-52 t_1 v_1^{11}+574 t_1^2 v_1^{10}-1573 t_1^3 v_1^9-22 t_2 v_1^9+1303 t_1^4 v_1^8+348 t_1 t_2 v_1^8-196 t_1 v_2 v_1^8-275 t_1^5 v_1^7-915 t_1^2 t_2 v_1^7+1837 t_1^2 v_2 v_1^7+6 t_1^6 v_1^6+43 t_2^2 v_1^6+530 t_1^3 t_2 v_1^6-4192 t_1^3 v_2 v_1^6-66 t_2 v_2 v_1^6-124 t_1 t_2^2 v_1^5-250 t_1 v_2^2 v_1^5-45 t_1^4 t_2 v_1^5-4 t_3 v_1^5+2933 t_1^4 v_2 v_1^5+864 t_1 t_2 v_2 v_1^5+39 t_1^2 t_2^2 v_1^4+1811 t_1^2 v_2^2 v_1^4+22 t_1 t_3 v_1^4-551 t_1^5 v_2 v_1^4-1809 t_1^2 t_2 v_2 v_1^4-44 t_1 v_3 v_1^4-2 t_2^3 v_1^3-3001 t_1^3 v_2^2 v_1^3-58 t_2 v_2^2 v_1^3-9 t_1^2 t_3 v_1^3+12 t_1^6 v_2 v_1^3+75 t_2^2 v_2 v_1^3+815 t_1^3 t_2 v_2 v_1^3+254 t_1^2 v_3 v_1^3-104 t_1 v_2^3 v_1^2+1543 t_1^4 v_2^2 v_1^2+528 t_1 t_2 v_2^2 v_1^2+3 t_2 t_3 v_1^2-153 t_1 t_2^2 v_2 v_1^2-60 t_1^4 t_2 v_2 v_1^2-6 t_3 v_2 v_1^2-382 t_1^3 v_3 v_1^2-12 t_2 v_3 v_1^2+456 t_1^2 v_2^3 v_1-226 t_1^5 v_2^2 v_1-714 t_1^2 t_2 v_2^2 v_1+39 t_1^2 t_2^2 v_2 v_1+24 t_1 t_3 v_2 v_1+203 t_1^4 v_3 v_1+96 t_1 t_2 v_3 v_1-48 t_1 v_2 v_3 v_1-260 t_1^3 v_2^3-8 t_2 v_2^3+4 t_1^6 v_2^2+18 t_2^2 v_2^2+140 t_1^3 t_2 v_2^2-2 t_2^3 v_2-6 t_1^2 t_3 v_2-28 t_1^5 v_3-84 t_1^2 t_2 v_3+120 t_1^2 v_2 v_3\right) d^{24}+\left(78 t_1 v_1^{12}-1039 t_1^2 v_1^{11}+3506 t_1^3 v_1^{10}+34 t_2 v_1^{10}-3861 t_1^4 v_1^9-639 t_1 t_2 v_1^9+320 t_1 v_2 v_1^9+1303 t_1^5 v_1^8+2222 t_1^2 t_2 v_1^8-3704 t_1^2 v_2 v_1^8-88 t_1^6 v_1^7-88 t_2^2 v_1^7-1927 t_1^3 t_2 v_1^7+10612 t_1^3 v_2 v_1^7+116 t_2 v_2 v_1^7+348 t_1 t_2^2 v_1^6+460 t_1 v_2^2 v_1^6+348 t_1^4 t_2 v_1^6+6 t_3 v_1^6-9900 t_1^4 v_2 v_1^6-1800 t_1 t_2 v_2 v_1^6-210 t_1^2 t_2^2 v_1^5-4325 t_1^2 v_2^2 v_1^5-4 t_1^5 t_2 v_1^5-45 t_1 t_3 v_1^5+2933 t_1^5 v_2 v_1^5+5150 t_1^2 t_2 v_2 v_1^5+72 t_1 v_3 v_1^5+10 t_2^3 v_1^4+10 t_1^3 t_2^2 v_1^4+9596 t_1^3 v_2^2 v_1^4+128 t_2 v_2^2 v_1^4+39 t_1^2 t_3 v_1^4-190 t_1^6 v_2 v_1^4-190 t_2^2 v_2 v_1^4-3638 t_1^3 t_2 v_2 v_1^4-582 t_1^2 v_3 v_1^4-2 t_1 t_2^3 v_1^3+240 t_1 v_2^3 v_1^3-6707 t_1^4 v_2^2 v_1^3-1404 t_1 t_2 v_2^2 v_1^3-2 t_1^3 t_3 v_1^3-9 t_2 t_3 v_1^3+552 t_1 t_2^2 v_2 v_1^3+552 t_1^4 t_2 v_2 v_1^3+12 t_3 v_2 v_1^3+1144 t_1^3 v_3 v_1^3+24 t_2 v_3 v_1^3-1564 t_1^2 v_2^3 v_1^2+1543 t_1^5 v_2^2 v_1^2+2819 t_1^2 t_2 v_2^2 v_1^2+3 t_1 t_2 t_3 v_1^2-252 t_1^2 t_2^2 v_2 v_1^2-6 t_1^5 t_2 v_2 v_1^2-60 t_1 t_3 v_2 v_1^2-791 t_1^4 v_3 v_1^2-216 t_1 t_2 v_3 v_1^2+96 t_1 v_2 v_3 v_1^2+2060 t_1^3 v_2^3 v_1+40 t_2 v_2^3 v_1-89 t_1^6 v_2^2 v_1-89 t_2^2 v_2^2 v_1-1318 t_1^3 t_2 v_2^2 v_1+11 t_2^3 v_2 v_1+11 t_1^3 t_2^2 v_2 v_1+39 t_1^2 t_3 v_2 v_1+203 t_1^5 v_3 v_1+406 t_1^2 t_2 v_3 v_1-520 t_1^2 v_2 v_3 v_1+16 t_1 v_2^4-523 t_1^4 v_2^3-144 t_1 t_2 v_2^3+102 t_1 t_2^2 v_2^2+102 t_1^4 t_2 v_2^2+4 t_3 v_2^2-2 t_1 t_2^3 v_2-2 t_1^3 t_3 v_2-6 t_2 t_3 v_2-14 t_1^6 v_3-14 t_2^2 v_3-140 t_1^3 t_2 v_3+424 t_1^3 v_2 v_3+16 t_2 v_2 v_3\right) d^{26}+\left(-118 t_1 v_1^{13}+1840 t_1^2 v_1^{12}-7546 t_1^3 v_1^{11}-52 t_2 v_1^{11}+10643 t_1^4 v_1^{10}+1170 t_1 t_2 v_1^{10}-528 t_1 v_2 v_1^{10}-5156 t_1^5 v_1^9-5067 t_1^2 t_2 v_1^9+7198 t_1^2 v_2 v_1^9+680 t_1^6 v_1^8+169 t_2^2 v_1^8+6131 t_1^3 t_2 v_1^8-25549 t_1^3 v_2 v_1^8-196 t_2 v_2 v_1^8-10 t_1^7 v_1^7-915 t_1 t_2^2 v_1^7-860 t_1 v_2^2 v_1^7-1927 t_1^4 t_2 v_1^7-10 t_3 v_1^7+30921 t_1^4 v_2 v_1^7+3740 t_1 t_2 v_2 v_1^7+868 t_1^2 t_2^2 v_1^6+9631 t_1^2 v_2^2 v_1^6+90 t_1^5 t_2 v_1^6+90 t_1 t_3 v_1^6-13042 t_1^5 v_2 v_1^6-13440 t_1^2 t_2 v_2 v_1^6-120 t_1 v_3 v_1^6-34 t_2^3 v_1^5-124 t_1^3 t_2^2 v_1^5-27679 t_1^3 v_2^2 v_1^5-250 t_2 v_2^2 v_1^5-124 t_1^2 t_3 v_1^5+1586 t_1^6 v_2 v_1^5+420 t_2^2 v_2 v_1^5+13555 t_1^3 t_2 v_2 v_1^5+1202 t_1^2 v_3 v_1^5+22 t_1 t_2^3 v_1^4-568 t_1 v_2^3 v_1^4+26363 t_1^4 v_2^2 v_1^4+3680 t_1 t_2 v_2^2 v_1^4+22 t_1^3 t_3 v_1^4+22 t_2 t_3 v_1^4-24 t_1^7 v_2 v_1^4-1809 t_1 t_2^2 v_2 v_1^4-3638 t_1^4 t_2 v_2 v_1^4-24 t_3 v_2 v_1^4-3170 t_1^3 v_3 v_1^4-44 t_2 v_3 v_1^4+4508 t_1^2 v_2^3 v_1^3-8671 t_1^5 v_2^2 v_1^3-9531 t_1^2 t_2 v_2^2 v_1^3-21 t_1 t_2 t_3 v_1^3+1311 t_1^2 t_2^2 v_2 v_1^3+156 t_1^5 t_2 v_2 v_1^3+156 t_1 t_3 v_2 v_1^3+2906 t_1^4 v_3 v_1^3+520 t_1 t_2 v_3 v_1^3-208 t_1 v_2 v_3 v_1^3-8920 t_1^3 v_2^3 v_1^2-104 t_2 v_2^3 v_1^2+879 t_1^6 v_2^2 v_1^2+257 t_2^2 v_2^2 v_1^2+6898 t_1^3 t_2 v_2^2 v_1^2-43 t_2^3 v_2 v_1^2-153 t_1^3 t_2^2 v_2 v_1^2-153 t_1^2 t_3 v_2 v_1^2-1001 t_1^5 v_3 v_1^2-1242 t_1^2 t_2 v_3 v_1^2+1416 t_1^2 v_2 v_3 v_1^2-104 t_1 v_2^4 v_1+5103 t_1^4 v_2^3 v_1+920 t_1 t_2 v_2^3 v_1-14 t_1^7 v_2^2 v_1-714 t_1 t_2^2 v_2^2 v_1-1318 t_1^4 t_2 v_2^2 v_1-14 t_3 v_2^2 v_1+24 t_1 t_2^3 v_2 v_1+24 t_1^3 t_3 v_2 v_1+24 t_2 t_3 v_2 v_1+126 t_1^6 v_3 v_1+46 t_2^2 v_3 v_1+896 t_1^3 t_2 v_3 v_1-2400 t_1^3 v_2 v_3 v_1-48 t_2 v_2 v_3 v_1+240 t_1^2 v_2^4-654 t_1^5 v_2^3-780 t_1^2 t_2 v_2^3+210 t_1^2 t_2^2 v_2^2+36 t_1^5 t_2 v_2^2+36 t_1 t_3 v_2^2-12 t_1 t_2 t_3 v_2-4 t_1^7 v_3-84 t_1 t_2^2 v_3-48 t_1 v_2^2 v_3-140 t_1^4 t_2 v_3-4 t_3 v_3+918 t_1^4 v_2 v_3+240 t_1 t_2 v_2 v_3\right) d^{28}+O\left(d^{30}\right)$

Another example where Method II excels is computing p-series, $$[p](x)=x\Fplus\cdots\Fplus x \quad\text{($p$ times).} $$

Let’s move to the prime p=3, switch to Araki generators, and compute the p-series by running BPSumToOrder[100,1][x,x,x] or equivalently

p=3;gen=Araki;
pSeriesBP[200,1][x]

which outputs

$3 x+d^4 v_1 x^3+\frac{9}{8} d^8 v_1^2 x^5+\frac{105}{64} d^{12} v_1^3 x^7+\frac{1377}{512} d^{16} v_1^4 x^9+\frac{3985389 d^{20} v_1^5 x^{11}}{839680}+\frac{59092773 d^{24} v_1^6 x^{13}}{6717440}+\frac{907229781 d^{28} v_1^7 x^{15}}{53739520}+\frac{2859206553 d^{32} v_1^8 x^{17}}{85983232}+\frac{47125533252921 d^{36} v_1^9 x^{19}}{705062502400}+\frac{770136329076849 d^{40} v_1^{10} x^{21}}{5640500019200}+\frac{12753311865572673 d^{44} v_1^{11} x^{23}}{45124000153600}+\frac{213532277359138857 d^{48} v_1^{12} x^{25}}{360992001228800}+\frac{739786458803938484949 d^{52} v_1^{13} x^{27}}{592026882015232000}+\frac{160172528275426594739975069765697 d^{56} v_1^{14} x^{29}}{60194116033843632543723683840}+\frac{13738036759250152674233106309048117 d^{60} v_1^{15} x^{31}}{2407764641353745301748947353600}+\frac{1185477657285968285516675671657275153 d^{64} v_1^{16} x^{33}}{96310585654149812069957894144000}+\frac{4216662079836460024248686951104873563261 d^{68} v_1^{17} x^{35}}{157949360472805691794730946396160000}+\frac{1470231164061376655327743560589353107201 7 d^{72} v_1^{18} x^{37}}{252718976756489106871569514233856000}+\frac{5148145496922714126297004923420454120958 1 d^{76} v_1^{19} x^{39}}{404350362810382570994511222774169600}+\frac{2261986546810199726838133196225415553541 5917 d^{80} v_1^{20} x^{41}}{80870072562076514198902244554833920000}+\frac{81780367256576925922593959120230527044 587607049 d^{84} v_1^{21} x^{43}}{132626919001805483286199681069927628800000}+\frac{1446850754068022215344154197219899 554433739571041 d^{88} v_1^{22} x^{45}}{1061015352014443866289597448559421030400000}+\frac{513427611175034102849726177908042 2333084076800477 d^{92} v_1^{23} x^{47}}{1697624563223110186063355917695073648640000}+\frac{456690167246529306226420669455873 418326343080577113 d^{96} v_1^{24} x^{49}}{67904982528924407442534236707802945945600000}+\frac{16695656116818900385200574489149 14396263990024439943141 d^{100} v_1^{25} x^{51}}{111364171347436028205756148200796831350784000000}+\frac{7278171017858302046215714122 99085854513880916800080117 d^{104} v_1^{26} x^{53}}{21729594409255810381610955746496942702592000000}+\frac{16981424497727856342183661748 3275107836618284414581196923503600973849 d^{108} v_1^{27} x^{55}}{2264582253184894429713717786111977316728786330337673216000000}+\frac{609463945405606 309062035428984430796639924562118998968145181222363661 d^{112} v_1^{28} x^{57}}{3623331605095831087541948457779163706766058128540277145600000}+\frac{112304392105755 24257619419784462139227943168786897763521612717937525008269 d^{116} v_1^{29} x^{59}}{29711319161785814917843977353789142395481676654030272593920000000}+\frac{20223239100 3004256418106108936578268353191736402021774131378054679681296901 d^{120} v_1^{30} x^{61}}{237690553294286519342751818830313139163853413232242180751360000000}+\frac{3647430129 095356423427669614071855519455147270410494624615116571869971787637 d^{124} v_1^{31} x^{63}}{1901524426354292154742014550642505113310827305857937446010880000000}+\frac{658814406 73225574039174522467643717638054608719738905973342309633526955084893 d^{128} v_1^{32} x^{65}}{15212195410834337237936116405140040906486618446863499568087040000000}+\frac{24428359 3110532235430591083138353368072170191211763203057563338040691731389194841 d^{132} v_1^{33} x^{67}}{24948000473768313070215230904429667086638054252856139291662745600000000}+\frac{88484 5995963904206064729063646734589517774341598365248314009675623765925443165277 d^{136} v_1^{34} x^{69}}{39916800758029300912344369447087467338620886804569822866660392960000000}+\frac{16045 192773117109726584378363098532358833777543403514716199218943853422453408308269 d^{140} v_1^{35} x^{71}}{319334406064234407298754955576699738708967094436558582933283143680000000}+\frac{1456 453195021108204728173502218102229770819846447193787044959616396498480278838048777 d^{144} v_1^{36} x^{73}}{12773376242569376291950198223067989548358683777462343317331325747200000000}+\frac{54 26367396518161843797379589227478297139341542363828872904806221441128726975366868393589 d^{148} v_1^{37} x^{75}}{20948337037813777118798325085831502859308241395038243040423374225408000000000}+\frac {19744685090892984759126199567593621261169588013339393380889587929721153931776974856111689 d^{152} v_1^{38} x^{77}}{33517339260502043390077320137330404574893186232061188864677398760652800000000}+\frac {71915147210074396046915779772281737879438554534983392433332150039293487288889386012014757 d^{156} v_1^{39} x^{79}}{53627742816803269424123712219728647319829097971297902183483838017044480000000}+\frac {1041288281714983914766668258702159641490110155511878904197189932064510783560704648826739758 9480547838193 d^{160} v_1^{40} x^{81}}{340786317290986485979421938267367134157520918135525516257174664404988360344050073600 0000000}+\frac{87752065535532880943076250832437715866958564012324142013399510533865924472306 769046530973212105631635587260769608465831332732347491987132461 d^{164} v_1^{41} x^{83}}{125928066598016465097161907674065208813049235901050573621694815453786845343621449192 72854388317206040411595182272675840000000000}+\frac{8568826145046047558402795865834205985031 02241718852724826297212188981782900485928241057368036142983694657943113184457112881241409982 5492607 d^{168} v_1^{42} x^{85}}{538729696676006267795345059568193406686841650913585341697090119588392921256134542000 977727842447317236859686942146560000000000}+\frac{117133477908259115379941932535493048361648 59109051322653254889595056401810987676647536885301382872804284820020670459350250112306194127 86801693 d^{172} v_1^{43} x^{87}}{322375850490922150648734483645606934561406043906689468471538727561694324079670909933 38507234092047463453683666618050150400000000}+\frac{5355693740810710185030318924940884235176 13398608111975469092173821135094687728177240210752316090212126710917046369227231039476229290 124816503517 d^{176} v_1^{44} x^{89}}{644751700981844301297468967291213869122812087813378936943077455123388648159341819866 7701446818409492690736733323610030080000000000}+\frac{20094906010247903986351432695578305362 76423709841669066835481452339649603074383819927743569717208897889050401197952341205794074775 433363983247998969 d^{180} v_1^{45} x^{91}}{105739278961022465412784910635759074536141182401394145658664702640235738298132058458 15030372782191568012808242650720449331200000000000}+\frac{3680528162965102053300275006483385 90686323659907941865331391711454413174983366957752849896087110842649569363174372010952011952 65298345729650995591953 d^{184} v_1^{46} x^{93}}{845914231688179723302279285086072596289129459211153165269317621121885906385056467665 20242982257532544102465941205763594649600000000000}+\frac{6745731712789738365297366638136039 85590824776936685125595935002591190118923180509191599401262560574536351281068756584622564570 150345440659156401962561 d^{188} v_1^{47} x^{95}}{676731385350543778641823428068858077031303567368922532215454096897508725108045174132 161943858060260352819727529646108757196800000000000}+\frac{989738597981512152257360525546172 68023790878190119292419290707626332114912048284091024976212148802044506375080702777394226982 137748619674310517670781 d^{192} v_1^{48} x^{97}}{433108086624348018330766993964069169300034283116110420617890622014405584069148911444 58364406915856662580462561897350960460595200000000}+\frac{4654327096445249523377958792291808 39166079303880477547210914197371318208021628369970607327058427044626278609226461789832747519 81881010922824510752019859221 d^{196} v_1^{49} x^{99}}{887871577579913437578072337626341797065070280388026362266675775129531447341755268461 3964703417750615828994825188956946894422016000000000000}+O\left(d^{200}\right)$

How do Methods I and II compare in performance? Here is a logarithmic plot of their runtimes when computing the p-series of $\mathrm{BP}\langle1\rangle$ at p=2 to various precisions, with Method I in blue and Method II in orange. As you can see, Method II scales far better. It should be said though that this is a particularly favorable match for Method II: the smaller k is, the better Method II handles $\mathrm{BP}\langle k\rangle$ since this makes the sum on the RHS of Theorem 4.3.9 small. On the other hand, Method II does even better at larger primes.

I was thrilled when I first tried Method II. It feels like a battering ram – sluggish to start, it crushes through high precision calculations.

In general, the code’s ability to compute FGL’s exceeds MathJax’s ability to display them in a browser.

Don’t have Mathematica? Get hold of a Raspberry Pi (which includes it).

p-typical Formal Group Law Lookup

For the prime p= , in terms of the $v_i$’s, the $p$-typical formula group law looks like

$\hspace{-10pt}x+_Fy=$

$(x+y) d-x y v_1 d^2+\left(x^2 y v_1^2+x y^2 v_1^2\right) d^3+\left(x^2 y^2 \left(-4 v_1^3-3 v_2\right)+x^3 y \left(-2 v_1^3-2 v_2\right)+x y^3 \left(-2 v_1^3-2 v_2\right)\right) d^4+\left(x^4 y \left(3 v_1^4+4 v_1 v_2\right)+x y^4 \left(3 v_1^4+4 v_1 v_2\right)+x^3 y^2 \left(10 v_1^4+11 v_1 v_2\right)+x^2 y^3 \left(10 v_1^4+11 v_1 v_2\right)\right) d^5+\left(x^3 y^3 \left(-34 v_1^5-43 v_1^2 v_2\right)+x^4 y^2 \left(-21 v_1^5-28 v_1^2 v_2\right)+x^2 y^4 \left(-21 v_1^5-28 v_1^2 v_2\right)+x^5 y \left(-4 v_1^5-6 v_1^2 v_2\right)+x y^5 \left(-4 v_1^5-6 v_1^2 v_2\right)\right) d^6+\left(x^6 y \left(6 v_1^6+12 v_1^3 v_2+4 v_2^2\right)+x y^6 \left(6 v_1^6+12 v_1^3 v_2+4 v_2^2\right)+x^5 y^2 \left(43 v_1^6+75 v_1^3 v_2+18 v_2^2\right)+x^2 y^5 \left(43 v_1^6+75 v_1^3 v_2+18 v_2^2\right)+x^4 y^3 \left(101 v_1^6+164 v_1^3 v_2+34 v_2^2\right)+x^3 y^4 \left(101 v_1^6+164 v_1^3 v_2+34 v_2^2\right)\right) d^7+\left(x^4 y^4 \left(-394 v_1^7-769 v_1^4 v_2-302 v_1 v_2^2-35 v_3\right)+x^5 y^3 \left(-275 v_1^7-551 v_1^4 v_2-226 v_1 v_2^2-28 v_3\right)+x^3 y^5 \left(-275 v_1^7-551 v_1^4 v_2-226 v_1 v_2^2-28 v_3\right)+x^6 y^2 \left(-88 v_1^7-190 v_1^4 v_2-89 v_1 v_2^2-14 v_3\right)+x^2 y^6 \left(-88 v_1^7-190 v_1^4 v_2-89 v_1 v_2^2-14 v_3\right)+x^7 y \left(-10 v_1^7-24 v_1^4 v_2-14 v_1 v_2^2-4 v_3\right)+x y^7 \left(-10 v_1^7-24 v_1^4 v_2-14 v_1 v_2^2-4 v_3\right)\right) d^8+\left(x^8 y \left(15 v_1^8+40 v_1^5 v_2+28 v_1^2 v_2^2+8 v_1 v_3\right)+x y^8 \left(15 v_1^8+40 v_1^5 v_2+28 v_1^2 v_2^2+8 v_1 v_3\right)+x^7 y^2 \left(169 v_1^8+420 v_1^5 v_2+257 v_1^2 v_2^2+46 v_1 v_3\right)+x^2 y^7 \left(169 v_1^8+420 v_1^5 v_2+257 v_1^2 v_2^2+46 v_1 v_3\right)+x^6 y^3 \left(680 v_1^8+1586 v_1^5 v_2+879 v_1^2 v_2^2+126 v_1 v_3\right)+x^3 y^6 \left(680 v_1^8+1586 v_1^5 v_2+879 v_1^2 v_2^2+126 v_1 v_3\right)+x^5 y^4 \left(1303 v_1^8+2933 v_1^5 v_2+1543 v_1^2 v_2^2+203 v_1 v_3\right)+x^4 y^5 \left(1303 v_1^8+2933 v_1^5 v_2+1543 v_1^2 v_2^2+203 v_1 v_3\right)\right) d^9+\left(x^5 y^5 \left(-5156 v_1^9-13042 v_1^6 v_2-8671 v_1^3 v_2^2-654 v_2^3-1001 v_1^2 v_3\right)+x^6 y^4 \left(-3861 v_1^9-9900 v_1^6 v_2-6707 v_1^3 v_2^2-523 v_2^3-791 v_1^2 v_3\right)+x^4 y^6 \left(-3861 v_1^9-9900 v_1^6 v_2-6707 v_1^3 v_2^2-523 v_2^3-791 v_1^2 v_3\right)+x^7 y^3 \left(-1573 v_1^9-4192 v_1^6 v_2-3001 v_1^3 v_2^2-260 v_2^3-382 v_1^2 v_3\right)+x^3 y^7 \left(-1573 v_1^9-4192 v_1^6 v_2-3001 v_1^3 v_2^2-260 v_2^3-382 v_1^2 v_3\right)+x^8 y^2 \left(-312 v_1^9-880 v_1^6 v_2-688 v_1^3 v_2^2-72 v_2^3-104 v_1^2 v_3\right)+x^2 y^8 \left(-312 v_1^9-880 v_1^6 v_2-688 v_1^3 v_2^2-72 v_2^3-104 v_1^2 v_3\right)+x^9 y \left(-22 v_1^9-66 v_1^6 v_2-58 v_1^3 v_2^2-8 v_2^3-12 v_1^2 v_3\right)+x y^9 \left(-22 v_1^9-66 v_1^6 v_2-58 v_1^3 v_2^2-8 v_2^3-12 v_1^2 v_3\right)\right) d^{10}+\left(x^{10} y \left(34 v_1^{10}+116 v_1^7 v_2+128 v_1^4 v_2^2+40 v_1 v_2^3+24 v_1^3 v_3+16 v_2 v_3\right)+x y^{10} \left(34 v_1^{10}+116 v_1^7 v_2+128 v_1^4 v_2^2+40 v_1 v_2^3+24 v_1^3 v_3+16 v_2 v_3\right)+x^9 y^2 \left(574 v_1^{10}+1837 v_1^7 v_2+1811 v_1^4 v_2^2+456 v_1 v_2^3+254 v_1^3 v_3+120 v_2 v_3\right)+x^2 y^9 \left(574 v_1^{10}+1837 v_1^7 v_2+1811 v_1^4 v_2^2+456 v_1 v_2^3+254 v_1^3 v_3+120 v_2 v_3\right)+x^8 y^3 \left(3506 v_1^{10}+10612 v_1^7 v_2+9596 v_1^4 v_2^2+2060 v_1 v_2^3+1144 v_1^3 v_3+424 v_2 v_3\right)+x^3 y^8 \left(3506 v_1^{10}+10612 v_1^7 v_2+9596 v_1^4 v_2^2+2060 v_1 v_2^3+1144 v_1^3 v_3+424 v_2 v_3\right)+x^7 y^4 \left(10643 v_1^{10}+30921 v_1^7 v_2+26363 v_1^4 v_2^2+5103 v_1 v_2^3+2906 v_1^3 v_3+918 v_2 v_3\right)+x^4 y^7 \left(10643 v_1^{10}+30921 v_1^7 v_2+26363 v_1^4 v_2^2+5103 v_1 v_2^3+2906 v_1^3 v_3+918 v_2 v_3\right)+x^6 y^5 \left(18115 v_1^{10}+51502 v_1^7 v_2+42619 v_1^4 v_2^2+7847 v_1 v_2^3+4557 v_1^3 v_3+1330 v_2 v_3\right)+x^5 y^6 \left(18115 v_1^{10}+51502 v_1^7 v_2+42619 v_1^4 v_2^2+7847 v_1 v_2^3+4557 v_1^3 v_3+1330 v_2 v_3\right)\right) d^{11}+\left(x^6 y^6 \left(-73850 v_1^{11}-231765 v_1^8 v_2-226598 v_1^5 v_2^2-62707 v_1^2 v_2^3-22715 v_1^4 v_3-13685 v_1 v_2 v_3\right)+x^7 y^5 \left(-58002 v_1^{11}-183665 v_1^8 v_2-181684 v_1^5 v_2^2-51133 v_1^2 v_2^3-18408 v_1^4 v_3-11384 v_1 v_2 v_3\right)+x^5 y^7 \left(-58002 v_1^{11}-183665 v_1^8 v_2-181684 v_1^5 v_2^2-51133 v_1^2 v_2^3-18408 v_1^4 v_3-11384 v_1 v_2 v_3\right)+x^8 y^4 \left(-27636 v_1^{11}-89822 v_1^8 v_2-91989 v_1^5 v_2^2-27241 v_1^2 v_2^3-9682 v_1^4 v_3-6470 v_1 v_2 v_3\right)+x^4 y^8 \left(-27636 v_1^{11}-89822 v_1^8 v_2-91989 v_1^5 v_2^2-27241 v_1^2 v_2^3-9682 v_1^4 v_3-6470 v_1 v_2 v_3\right)+x^9 y^3 \left(-7546 v_1^{11}-25549 v_1^8 v_2-27679 v_1^5 v_2^2-8920 v_1^2 v_2^3-3170 v_1^4 v_3-2400 v_1 v_2 v_3\right)+x^3 y^9 \left(-7546 v_1^{11}-25549 v_1^8 v_2-27679 v_1^5 v_2^2-8920 v_1^2 v_2^3-3170 v_1^4 v_3-2400 v_1 v_2 v_3\right)+x^{10} y^2 \left(-1039 v_1^{11}-3704 v_1^8 v_2-4325 v_1^5 v_2^2-1564 v_1^2 v_2^3-582 v_1^4 v_3-520 v_1 v_2 v_3\right)+x^2 y^{10} \left(-1039 v_1^{11}-3704 v_1^8 v_2-4325 v_1^5 v_2^2-1564 v_1^2 v_2^3-582 v_1^4 v_3-520 v_1 v_2 v_3\right)+x^{11} y \left(-52 v_1^{11}-196 v_1^8 v_2-250 v_1^5 v_2^2-104 v_1^2 v_2^3-44 v_1^4 v_3-48 v_1 v_2 v_3\right)+x y^{11} \left(-52 v_1^{11}-196 v_1^8 v_2-250 v_1^5 v_2^2-104 v_1^2 v_2^3-44 v_1^4 v_3-48 v_1 v_2 v_3\right)\right) d^{12}+\left(x^{12} y \left(78 v_1^{12}+320 v_1^9 v_2+460 v_1^6 v_2^2+240 v_1^3 v_2^3+16 v_2^4+72 v_1^5 v_3+96 v_1^2 v_2 v_3\right)+x y^{12} \left(78 v_1^{12}+320 v_1^9 v_2+460 v_1^6 v_2^2+240 v_1^3 v_2^3+16 v_2^4+72 v_1^5 v_3+96 v_1^2 v_2 v_3\right)+x^{11} y^2 \left(1840 v_1^{12}+7198 v_1^9 v_2+9631 v_1^6 v_2^2+4508 v_1^3 v_2^3+240 v_2^4+1202 v_1^5 v_3+1416 v_1^2 v_2 v_3\right)+x^2 y^{11} \left(1840 v_1^{12}+7198 v_1^9 v_2+9631 v_1^6 v_2^2+4508 v_1^3 v_2^3+240 v_2^4+1202 v_1^5 v_3+1416 v_1^2 v_2 v_3\right)+x^{10} y^3 \left(15709 v_1^{12}+58672 v_1^9 v_2+73615 v_1^6 v_2^2+31416 v_1^3 v_2^3+1400 v_2^4+7934 v_1^5 v_3+8352 v_1^2 v_2 v_3\right)+x^3 y^{10} \left(15709 v_1^{12}+58672 v_1^9 v_2+73615 v_1^6 v_2^2+31416 v_1^3 v_2^3+1400 v_2^4+7934 v_1^5 v_3+8352 v_1^2 v_2 v_3\right)+x^9 y^4 \left(68193 v_1^{12}+244934 v_1^9 v_2+291819 v_1^6 v_2^2+115864 v_1^3 v_2^3+4546 v_2^4+28937 v_1^5 v_3+27744 v_1^2 v_2 v_3\right)+x^4 y^9 \left(68193 v_1^{12}+244934 v_1^9 v_2+291819 v_1^6 v_2^2+115864 v_1^3 v_2^3+4546 v_2^4+28937 v_1^5 v_3+27744 v_1^2 v_2 v_3\right)+x^8 y^5 \left(172287 v_1^{12}+601728 v_1^9 v_2+691569 v_1^6 v_2^2+261417 v_1^3 v_2^3+9462 v_2^4+65784 v_1^5 v_3+58986 v_1^2 v_2 v_3\right)+x^5 y^8 \left(172287 v_1^{12}+601728 v_1^9 v_2+691569 v_1^6 v_2^2+261417 v_1^3 v_2^3+9462 v_2^4+65784 v_1^5 v_3+58986 v_1^2 v_2 v_3\right)+x^7 y^6 \left(270163 v_1^{12}+929923 v_1^9 v_2+1049281 v_1^6 v_2^2+386948 v_1^3 v_2^3+13470 v_2^4+98170 v_1^5 v_3+85011 v_1^2 v_2 v_3\right)+x^6 y^7 \left(270163 v_1^{12}+929923 v_1^9 v_2+1049281 v_1^6 v_2^2+386948 v_1^3 v_2^3+13470 v_2^4+98170 v_1^5 v_3+85011 v_1^2 v_2 v_3\right)\right) d^{13}+O[d]^{14}$
$(x+y) d+\left(-x^2 y v_1-x y^2 v_1\right) d^3+\left(x^4 y v_1^2+3 x^3 y^2 v_1^2+3 x^2 y^3 v_1^2+x y^4 v_1^2\right) d^5+\left(-x^6 y v_1^3-6 x^5 y^2 v_1^3-13 x^4 y^3 v_1^3-13 x^3 y^4 v_1^3-6 x^2 y^5 v_1^3-x y^6 v_1^3\right) d^7+\left(x^5 y^4 \left(52 v_1^4-42 v_2\right)+x^4 y^5 \left(52 v_1^4-42 v_2\right)+x^6 y^3 \left(27 v_1^4-28 v_2\right)+x^3 y^6 \left(27 v_1^4-28 v_2\right)+x^7 y^2 \left(6 v_1^4-12 v_2\right)+x^2 y^7 \left(6 v_1^4-12 v_2\right)-3 x^8 y v_2-3 x y^8 v_2\right) d^9+\left(45 x^9 y^2 v_1 v_2+45 x^2 y^9 v_1 v_2+x^{10} y \left(v_1^5+6 v_1 v_2\right)+x y^{10} \left(v_1^5+6 v_1 v_2\right)+x^8 y^3 \left(-27 v_1^5+163 v_1 v_2\right)+x^3 y^8 \left(-27 v_1^5+163 v_1 v_2\right)+x^7 y^4 \left(-106 v_1^5+362 v_1 v_2\right)+x^4 y^7 \left(-106 v_1^5+362 v_1 v_2\right)+x^6 y^5 \left(-192 v_1^5+532 v_1 v_2\right)+x^5 y^6 \left(-192 v_1^5+532 v_1 v_2\right)\right) d^{11}+\left(x^7 y^6 \left(484 v_1^6-5164 v_1^2 v_2\right)+x^6 y^7 \left(484 v_1^6-5164 v_1^2 v_2\right)+x^8 y^5 \left(246 v_1^6-3637 v_1^2 v_2\right)+x^5 y^8 \left(246 v_1^6-3637 v_1^2 v_2\right)+x^9 y^4 \left(30 v_1^6-1770 v_1^2 v_2\right)+x^4 y^9 \left(30 v_1^6-1770 v_1^2 v_2\right)+x^{10} y^3 \left(-31 v_1^6-568 v_1^2 v_2\right)+x^3 y^{10} \left(-31 v_1^6-568 v_1^2 v_2\right)+x^{11} y^2 \left(-15 v_1^6-108 v_1^2 v_2\right)+x^2 y^{11} \left(-15 v_1^6-108 v_1^2 v_2\right)+x^{12} y \left(-2 v_1^6-9 v_1^2 v_2\right)+x y^{12} \left(-2 v_1^6-9 v_1^2 v_2\right)\right) d^{13}+\left(x^{14} y \left(3 v_1^7+12 v_1^3 v_2\right)+x y^{14} \left(3 v_1^7+12 v_1^3 v_2\right)+x^{13} y^2 \left(42 v_1^7+210 v_1^3 v_2\right)+x^2 y^{13} \left(42 v_1^7+210 v_1^3 v_2\right)+x^{12} y^3 \left(226 v_1^7+1517 v_1^3 v_2\right)+x^3 y^{12} \left(226 v_1^7+1517 v_1^3 v_2\right)+x^{11} y^4 \left(655 v_1^7+6333 v_1^3 v_2\right)+x^4 y^{11} \left(655 v_1^7+6333 v_1^3 v_2\right)+x^{10} y^5 \left(1168 v_1^7+17350 v_1^3 v_2\right)+x^5 y^{10} \left(1168 v_1^7+17350 v_1^3 v_2\right)+x^9 y^6 \left(1412 v_1^7+33137 v_1^3 v_2\right)+x^6 y^9 \left(1412 v_1^7+33137 v_1^3 v_2\right)+x^8 y^7 \left(1370 v_1^7+45493 v_1^3 v_2\right)+x^7 y^8 \left(1370 v_1^7+45493 v_1^3 v_2\right)\right) d^{15}+\left(x^{16} y \left(-3 v_1^8-9 v_1^4 v_2+9 v_2^2\right)+x y^{16} \left(-3 v_1^8-9 v_1^4 v_2+9 v_2^2\right)+x^{15} y^2 \left(-72 v_1^8-288 v_1^4 v_2+108 v_2^2\right)+x^2 y^{15} \left(-72 v_1^8-288 v_1^4 v_2+108 v_2^2\right)+x^{14} y^3 \left(-613 v_1^8-3010 v_1^4 v_2+624 v_2^2\right)+x^3 y^{14} \left(-613 v_1^8-3010 v_1^4 v_2+624 v_2^2\right)+x^{13} y^4 \left(-2842 v_1^8-16940 v_1^4 v_2+2310 v_2^2\right)+x^4 y^{13} \left(-2842 v_1^8-16940 v_1^4 v_2+2310 v_2^2\right)+x^{12} y^5 \left(-8500 v_1^8-60962 v_1^4 v_2+6132 v_2^2\right)+x^5 y^{12} \left(-8500 v_1^8-60962 v_1^4 v_2+6132 v_2^2\right)+x^{11} y^6 \left(-17987 v_1^8-152080 v_1^4 v_2+12348 v_2^2\right)+x^6 y^{11} \left(-17987 v_1^8-152080 v_1^4 v_2+12348 v_2^2\right)+x^{10} y^7 \left(-28612 v_1^8-274530 v_1^4 v_2+19440 v_2^2\right)+x^7 y^{10} \left(-28612 v_1^8-274530 v_1^4 v_2+19440 v_2^2\right)+x^9 y^8 \left(-35675 v_1^8-366965 v_1^4 v_2+24309 v_2^2\right)+x^8 y^9 \left(-35675 v_1^8-366965 v_1^4 v_2+24309 v_2^2\right)\right) d^{17}+\left(x^{10} y^9 \left(388687 v_1^9+2747309 v_1^5 v_2-608258 v_1 v_2^2\right)+x^9 y^{10} \left(388687 v_1^9+2747309 v_1^5 v_2-608258 v_1 v_2^2\right)+x^{11} y^8 \left(306815 v_1^9+2099671 v_1^5 v_2-490014 v_1 v_2^2\right)+x^8 y^{11} \left(306815 v_1^9+2099671 v_1^5 v_2-490014 v_1 v_2^2\right)+x^{12} y^7 \left(189242 v_1^9+1216154 v_1^5 v_2-316276 v_1 v_2^2\right)+x^7 y^{12} \left(189242 v_1^9+1216154 v_1^5 v_2-316276 v_1 v_2^2\right)+x^{13} y^6 \left(89278 v_1^9+523992 v_1^5 v_2-161560 v_1 v_2^2\right)+x^6 y^{13} \left(89278 v_1^9+523992 v_1^5 v_2-161560 v_1 v_2^2\right)+x^{14} y^5 \left(31084 v_1^9+162382 v_1^5 v_2-63888 v_1 v_2^2\right)+x^5 y^{14} \left(31084 v_1^9+162382 v_1^5 v_2-63888 v_1 v_2^2\right)+x^{15} y^4 \left(7536 v_1^9+34080 v_1^5 v_2-18828 v_1 v_2^2\right)+x^4 y^{15} \left(7536 v_1^9+34080 v_1^5 v_2-18828 v_1 v_2^2\right)+x^{16} y^3 \left(1150 v_1^9+4305 v_1^5 v_2-3867 v_1 v_2^2\right)+x^3 y^{16} \left(1150 v_1^9+4305 v_1^5 v_2-3867 v_1 v_2^2\right)+x^{17} y^2 \left(90 v_1^9+243 v_1^5 v_2-486 v_1 v_2^2\right)+x^2 y^{17} \left(90 v_1^9+243 v_1^5 v_2-486 v_1 v_2^2\right)+x^{18} y \left(2 v_1^9-27 v_1 v_2^2\right)+x y^{18} \left(2 v_1^9-27 v_1 v_2^2\right)\right) d^{19}+\left(x^{20} y \left(15 v_1^6 v_2+54 v_1^2 v_2^2\right)+x y^{20} \left(15 v_1^6 v_2+54 v_1^2 v_2^2\right)+x^{19} y^2 \left(-75 v_1^{10}+60 v_1^6 v_2+1350 v_1^2 v_2^2\right)+x^2 y^{19} \left(-75 v_1^{10}+60 v_1^6 v_2+1350 v_1^2 v_2^2\right)+x^{18} y^3 \left(-1594 v_1^{10}-3264 v_1^6 v_2+14121 v_1^2 v_2^2\right)+x^3 y^{18} \left(-1594 v_1^{10}-3264 v_1^6 v_2+14121 v_1^2 v_2^2\right)+x^{17} y^4 \left(-14730 v_1^{10}-47088 v_1^6 v_2+87561 v_1^2 v_2^2\right)+x^4 y^{17} \left(-14730 v_1^{10}-47088 v_1^6 v_2+87561 v_1^2 v_2^2\right)+x^{16} y^5 \left(-81106 v_1^{10}-318756 v_1^6 v_2+370854 v_1^2 v_2^2\right)+x^5 y^{16} \left(-81106 v_1^{10}-318756 v_1^6 v_2+370854 v_1^2 v_2^2\right)+x^{15} y^6 \left(-302978 v_1^{10}-1358816 v_1^6 v_2+1155684 v_1^2 v_2^2\right)+x^6 y^{15} \left(-302978 v_1^{10}-1358816 v_1^6 v_2+1155684 v_1^2 v_2^2\right)+x^{14} y^7 \left(-822816 v_1^{10}-4051162 v_1^6 v_2+2767888 v_1^2 v_2^2\right)+x^7 y^{14} \left(-822816 v_1^{10}-4051162 v_1^6 v_2+2767888 v_1^2 v_2^2\right)+x^{13} y^8 \left(-1692592 v_1^{10}-8902137 v_1^6 v_2+5232925 v_1^2 v_2^2\right)+x^8 y^{13} \left(-1692592 v_1^{10}-8902137 v_1^6 v_2+5232925 v_1^2 v_2^2\right)+x^{12} y^9 \left(-2704615 v_1^{10}-14847394 v_1^6 v_2+7938873 v_1^2 v_2^2\right)+x^9 y^{12} \left(-2704615 v_1^{10}-14847394 v_1^6 v_2+7938873 v_1^2 v_2^2\right)+x^{11} y^{10} \left(-3407106 v_1^{10}-19104154 v_1^6 v_2+9757046 v_1^2 v_2^2\right)+x^{10} y^{11} \left(-3407106 v_1^{10}-19104154 v_1^6 v_2+9757046 v_1^2 v_2^2\right)\right) d^{21}+\left(x^{12} y^{11} \left(26625777 v_1^{11}+121340063 v_1^7 v_2-127710264 v_1^3 v_2^2\right)+x^{11} y^{12} \left(26625777 v_1^{11}+121340063 v_1^7 v_2-127710264 v_1^3 v_2^2\right)+x^{13} y^{10} \left(21339672 v_1^{11}+95416130 v_1^7 v_2-105024048 v_1^3 v_2^2\right)+x^{10} y^{13} \left(21339672 v_1^{11}+95416130 v_1^7 v_2-105024048 v_1^3 v_2^2\right)+x^{14} y^9 \left(13632623 v_1^{11}+58574591 v_1^7 v_2-70778604 v_1^3 v_2^2\right)+x^9 y^{14} \left(13632623 v_1^{11}+58574591 v_1^7 v_2-70778604 v_1^3 v_2^2\right)+x^{15} y^8 \left(6860630 v_1^{11}+27617558 v_1^7 v_2-38799876 v_1^3 v_2^2\right)+x^8 y^{15} \left(6860630 v_1^{11}+27617558 v_1^7 v_2-38799876 v_1^3 v_2^2\right)+x^{16} y^7 \left(2665810 v_1^{11}+9701268 v_1^7 v_2-17086090 v_1^3 v_2^2\right)+x^7 y^{16} \left(2665810 v_1^{11}+9701268 v_1^7 v_2-17086090 v_1^3 v_2^2\right)+x^{17} y^6 \left(774174 v_1^{11}+2394027 v_1^7 v_2-5927094 v_1^3 v_2^2\right)+x^6 y^{17} \left(774174 v_1^{11}+2394027 v_1^7 v_2-5927094 v_1^3 v_2^2\right)+x^{18} y^5 \left(159130 v_1^{11}+360792 v_1^7 v_2-1571454 v_1^3 v_2^2\right)+x^5 y^{18} \left(159130 v_1^{11}+360792 v_1^7 v_2-1571454 v_1^3 v_2^2\right)+x^{19} y^4 \left(20915 v_1^{11}+16490 v_1^7 v_2-303600 v_1^3 v_2^2\right)+x^4 y^{19} \left(20915 v_1^{11}+16490 v_1^7 v_2-303600 v_1^3 v_2^2\right)+x^{20} y^3 \left(1368 v_1^{11}-4475 v_1^7 v_2-39462 v_1^3 v_2^2\right)+x^3 y^{20} \left(1368 v_1^{11}-4475 v_1^7 v_2-39462 v_1^3 v_2^2\right)+x^{21} y^2 \left(-792 v_1^7 v_2-2970 v_1^3 v_2^2\right)+x^2 y^{21} \left(-792 v_1^7 v_2-2970 v_1^3 v_2^2\right)+x^{22} y \left(-3 v_1^{11}-36 v_1^7 v_2-90 v_1^3 v_2^2\right)+x y^{22} \left(-3 v_1^{11}-36 v_1^7 v_2-90 v_1^3 v_2^2\right)\right) d^{23}+O[d]^{24}$
$(x+y) d+\left(-x^4 y v_1-2 x^3 y^2 v_1-2 x^2 y^3 v_1-x y^4 v_1\right) d^5+\left(x^8 y v_1^2+6 x^7 y^2 v_1^2+16 x^6 y^3 v_1^2+25 x^5 y^4 v_1^2+25 x^4 y^5 v_1^2+16 x^3 y^6 v_1^2+6 x^2 y^7 v_1^2+x y^8 v_1^2\right) d^9+\left(-x^{12} y v_1^3-12 x^{11} y^2 v_1^3-60 x^{10} y^3 v_1^3-175 x^9 y^4 v_1^3-340 x^8 y^5 v_1^3-468 x^7 y^6 v_1^3-468 x^6 y^7 v_1^3-340 x^5 y^8 v_1^3-175 x^4 y^9 v_1^3-60 x^3 y^{10} v_1^3-12 x^2 y^{11} v_1^3-x y^{12} v_1^3\right) d^{13}+\left(x^{16} y v_1^4+20 x^{15} y^2 v_1^4+160 x^{14} y^3 v_1^4+735 x^{13} y^4 v_1^4+2251 x^{12} y^5 v_1^4+4968 x^{11} y^6 v_1^4+8256 x^{10} y^7 v_1^4+10585 x^9 y^8 v_1^4+10585 x^8 y^9 v_1^4+8256 x^7 y^{10} v_1^4+4968 x^6 y^{11} v_1^4+2251 x^5 y^{12} v_1^4+735 x^4 y^{13} v_1^4+160 x^3 y^{14} v_1^4+20 x^2 y^{15} v_1^4+x y^{16} v_1^4\right) d^{17}+\left(-x^{20} y v_1^5-30 x^{19} y^2 v_1^5-350 x^{18} y^3 v_1^5-2310 x^{17} y^4 v_1^5-10105 x^{16} y^5 v_1^5-31912 x^{15} y^6 v_1^5-76596 x^{14} y^7 v_1^5-144348 x^{13} y^8 v_1^5-218026 x^{12} y^9 v_1^5-267186 x^{11} y^{10} v_1^5-267186 x^{10} y^{11} v_1^5-218026 x^9 y^{12} v_1^5-144348 x^8 y^{13} v_1^5-76596 x^7 y^{14} v_1^5-31912 x^6 y^{15} v_1^5-10105 x^5 y^{16} v_1^5-2310 x^4 y^{17} v_1^5-350 x^3 y^{18} v_1^5-30 x^2 y^{19} v_1^5-x y^{20} v_1^5\right) d^{21}+\left(x^{13} y^{12} \left(7037846 v_1^6-1040060 v_2\right)+x^{12} y^{13} \left(7037846 v_1^6-1040060 v_2\right)+x^{14} y^{11} \left(5922156 v_1^6-891480 v_2\right)+x^{11} y^{14} \left(5922156 v_1^6-891480 v_2\right)+x^{15} y^{10} \left(4178890 v_1^6-653752 v_2\right)+x^{10} y^{15} \left(4178890 v_1^6-653752 v_2\right)+x^{16} y^9 \left(2454920 v_1^6-408595 v_2\right)+x^9 y^{16} \left(2454920 v_1^6-408595 v_2\right)+x^{17} y^8 \left(1186548 v_1^6-216315 v_2\right)+x^8 y^{17} \left(1186548 v_1^6-216315 v_2\right)+x^{18} y^7 \left(463550 v_1^6-96140 v_2\right)+x^7 y^{18} \left(463550 v_1^6-96140 v_2\right)+x^{19} y^6 \left(142592 v_1^6-35420 v_2\right)+x^6 y^{19} \left(142592 v_1^6-35420 v_2\right)+x^{20} y^5 \left(33205 v_1^6-10626 v_2\right)+x^5 y^{20} \left(33205 v_1^6-10626 v_2\right)+x^{21} y^4 \left(5500 v_1^6-2530 v_2\right)+x^4 y^{21} \left(5500 v_1^6-2530 v_2\right)+x^{22} y^3 \left(580 v_1^6-460 v_2\right)+x^3 y^{22} \left(580 v_1^6-460 v_2\right)+x^{23} y^2 \left(30 v_1^6-60 v_2\right)+x^2 y^{23} \left(30 v_1^6-60 v_2\right)-5 x^{24} y v_2-5 x y^{24} v_2\right) d^{25}+\left(x^{28} y \left(v_1^7+10 v_1 v_2\right)+x y^{28} \left(v_1^7+10 v_1 v_2\right)+x^{27} y^2 \left(-14 v_1^7+210 v_1 v_2\right)+x^2 y^{27} \left(-14 v_1^7+210 v_1 v_2\right)+x^{26} y^3 \left(-704 v_1^7+2360 v_1 v_2\right)+x^3 y^{26} \left(-704 v_1^7+2360 v_1 v_2\right)+x^{25} y^4 \left(-10075 v_1^7+17875 v_1 v_2\right)+x^4 y^{25} \left(-10075 v_1^7+17875 v_1 v_2\right)+x^{24} y^5 \left(-83580 v_1^7+100001 v_1 v_2\right)+x^5 y^{24} \left(-83580 v_1^7+100001 v_1 v_2\right)+x^{23} y^6 \left(-476912 v_1^7+435404 v_1 v_2\right)+x^6 y^{23} \left(-476912 v_1^7+435404 v_1 v_2\right)+x^{22} y^7 \left(-2030448 v_1^7+1526556 v_1 v_2\right)+x^7 y^{22} \left(-2030448 v_1^7+1526556 v_1 v_2\right)+x^{21} y^8 \left(-6768685 v_1^7+4413079 v_1 v_2\right)+x^8 y^{21} \left(-6768685 v_1^7+4413079 v_1 v_2\right)+x^{20} y^9 \left(-18235685 v_1^7+10699876 v_1 v_2\right)+x^9 y^{20} \left(-18235685 v_1^7+10699876 v_1 v_2\right)+x^{19} y^{10} \left(-40583850 v_1^7+22032252 v_1 v_2\right)+x^{10} y^{19} \left(-40583850 v_1^7+22032252 v_1 v_2\right)+x^{18} y^{11} \left(-75775238 v_1^7+38886008 v_1 v_2\right)+x^{11} y^{18} \left(-75775238 v_1^7+38886008 v_1 v_2\right)+x^{17} y^{12} \left(-120005378 v_1^7+59224862 v_1 v_2\right)+x^{12} y^{17} \left(-120005378 v_1^7+59224862 v_1 v_2\right)+x^{16} y^{13} \left(-162416316 v_1^7+78205083 v_1 v_2\right)+x^{13} y^{16} \left(-162416316 v_1^7+78205083 v_1 v_2\right)+x^{15} y^{14} \left(-188735180 v_1^7+89801752 v_1 v_2\right)+x^{14} y^{15} \left(-188735180 v_1^7+89801752 v_1 v_2\right)\right) d^{29}+\left(x^{17} y^{16} \left(5082032154 v_1^8-5200107720 v_1^2 v_2\right)+x^{16} y^{17} \left(5082032154 v_1^8-5200107720 v_1^2 v_2\right)+x^{18} y^{15} \left(4448767094 v_1^8-4591688352 v_1^2 v_2\right)+x^{15} y^{18} \left(4448767094 v_1^8-4591688352 v_1^2 v_2\right)+x^{19} y^{14} \left(3403766940 v_1^8-3576153252 v_1^2 v_2\right)+x^{14} y^{19} \left(3403766940 v_1^8-3576153252 v_1^2 v_2\right)+x^{20} y^{13} \left(2268757917 v_1^8-2451145926 v_1^2 v_2\right)+x^{13} y^{20} \left(2268757917 v_1^8-2451145926 v_1^2 v_2\right)+x^{21} y^{12} \left(1310694438 v_1^8-1473376458 v_1^2 v_2\right)+x^{12} y^{21} \left(1310694438 v_1^8-1473376458 v_1^2 v_2\right)+x^{22} y^{11} \left(651497208 v_1^8-772881972 v_1^2 v_2\right)+x^{11} y^{22} \left(651497208 v_1^8-772881972 v_1^2 v_2\right)+x^{23} y^{10} \left(275822202 v_1^8-351476952 v_1^2 v_2\right)+x^{10} y^{23} \left(275822202 v_1^8-351476952 v_1^2 v_2\right)+x^{24} y^9 \left(98099560 v_1^8-137368626 v_1^2 v_2\right)+x^9 y^{24} \left(98099560 v_1^8-137368626 v_1^2 v_2\right)+x^{25} y^8 \left(28761070 v_1^8-45618625 v_1^2 v_2\right)+x^8 y^{25} \left(28761070 v_1^8-45618625 v_1^2 v_2\right)+x^{26} y^7 \left(6767592 v_1^8-12680988 v_1^2 v_2\right)+x^7 y^{26} \left(6767592 v_1^8-12680988 v_1^2 v_2\right)+x^{27} y^6 \left(1228164 v_1^8-2892092 v_1^2 v_2\right)+x^6 y^{27} \left(1228164 v_1^8-2892092 v_1^2 v_2\right)+x^{28} y^5 \left(160985 v_1^8-526426 v_1^2 v_2\right)+x^5 y^{28} \left(160985 v_1^8-526426 v_1^2 v_2\right)+x^{29} y^4 \left(13350 v_1^8-73500 v_1^2 v_2\right)+x^4 y^{29} \left(13350 v_1^8-73500 v_1^2 v_2\right)+x^{30} y^3 \left(440 v_1^8-7400 v_1^2 v_2\right)+x^3 y^{30} \left(440 v_1^8-7400 v_1^2 v_2\right)+x^{31} y^2 \left(-24 v_1^8-480 v_1^2 v_2\right)+x^2 y^{31} \left(-24 v_1^8-480 v_1^2 v_2\right)+x^{32} y \left(-2 v_1^8-15 v_1^2 v_2\right)+x y^{32} \left(-2 v_1^8-15 v_1^2 v_2\right)\right) d^{33}+\left(x^{36} y \left(3 v_1^9+20 v_1^3 v_2\right)+x y^{36} \left(3 v_1^9+20 v_1^3 v_2\right)+x^{35} y^2 \left(90 v_1^9+900 v_1^3 v_2\right)+x^2 y^{35} \left(90 v_1^9+900 v_1^3 v_2\right)+x^{34} y^3 \left(670 v_1^9+18200 v_1^3 v_2\right)+x^3 y^{34} \left(670 v_1^9+18200 v_1^3 v_2\right)+x^{33} y^4 \left(-7480 v_1^9+229075 v_1^3 v_2\right)+x^4 y^{33} \left(-7480 v_1^9+229075 v_1^3 v_2\right)+x^{32} y^5 \left(-210013 v_1^9+2040021 v_1^3 v_2\right)+x^5 y^{32} \left(-210013 v_1^9+2040021 v_1^3 v_2\right)+x^{31} y^6 \left(-2347776 v_1^9+13774464 v_1^3 v_2\right)+x^6 y^{31} \left(-2347776 v_1^9+13774464 v_1^3 v_2\right)+x^{30} y^7 \left(-17164524 v_1^9+73682400 v_1^3 v_2\right)+x^7 y^{30} \left(-17164524 v_1^9+73682400 v_1^3 v_2\right)+x^{29} y^8 \left(-93126045 v_1^9+321901575 v_1^3 v_2\right)+x^8 y^{29} \left(-93126045 v_1^9+321901575 v_1^3 v_2\right)+x^{28} y^9 \left(-398129180 v_1^9+1174371076 v_1^3 v_2\right)+x^9 y^{28} \left(-398129180 v_1^9+1174371076 v_1^3 v_2\right)+x^{27} y^{10} \left(-1390133088 v_1^9+3638242272 v_1^3 v_2\right)+x^{10} y^{27} \left(-1390133088 v_1^9+3638242272 v_1^3 v_2\right)+x^{26} y^{11} \left(-4060627464 v_1^9+9696014292 v_1^3 v_2\right)+x^{11} y^{26} \left(-4060627464 v_1^9+9696014292 v_1^3 v_2\right)+x^{25} y^{12} \left(-10094057493 v_1^9+22453931955 v_1^3 v_2\right)+x^{12} y^{25} \left(-10094057493 v_1^9+22453931955 v_1^3 v_2\right)+x^{24} y^{13} \left(-21625097192 v_1^9+45544056176 v_1^3 v_2\right)+x^{13} y^{24} \left(-21625097192 v_1^9+45544056176 v_1^3 v_2\right)+x^{23} y^{14} \left(-40307191452 v_1^9+81416189052 v_1^3 v_2\right)+x^{14} y^{23} \left(-40307191452 v_1^9+81416189052 v_1^3 v_2\right)+x^{22} y^{15} \left(-65830199944 v_1^9+128890913572 v_1^3 v_2\right)+x^{15} y^{22} \left(-65830199944 v_1^9+128890913572 v_1^3 v_2\right)+x^{21} y^{16} \left(-94702748416 v_1^9+181362401170 v_1^3 v_2\right)+x^{16} y^{21} \left(-94702748416 v_1^9+181362401170 v_1^3 v_2\right)+x^{20} y^{17} \left(-120448687068 v_1^9+227415961776 v_1^3 v_2\right)+x^{17} y^{20} \left(-120448687068 v_1^9+227415961776 v_1^3 v_2\right)+x^{19} y^{18} \left(-135759799484 v_1^9+254552594852 v_1^3 v_2\right)+x^{18} y^{19} \left(-135759799484 v_1^9+254552594852 v_1^3 v_2\right)\right) d^{37}+O[d]^{38}$
$(x+y) d+\left(-x^6 y v_1-3 x^5 y^2 v_1-5 x^4 y^3 v_1-5 x^3 y^4 v_1-3 x^2 y^5 v_1-x y^6 v_1\right) d^7+\left(x^{12} y v_1^2+9 x^{11} y^2 v_1^2+38 x^{10} y^3 v_1^2+100 x^9 y^4 v_1^2+183 x^8 y^5 v_1^2+245 x^7 y^6 v_1^2+245 x^6 y^7 v_1^2+183 x^5 y^8 v_1^2+100 x^4 y^9 v_1^2+38 x^3 y^{10} v_1^2+9 x^2 y^{11} v_1^2+x y^{12} v_1^2\right) d^{13}+\left(-x^{18} y v_1^3-18 x^{17} y^2 v_1^3-140 x^{16} y^3 v_1^3-660 x^{15} y^4 v_1^3-2163 x^{14} y^5 v_1^3-5292 x^{13} y^6 v_1^3-10073 x^{12} y^7 v_1^3-15291 x^{11} y^8 v_1^3-18778 x^{10} y^9 v_1^3-18778 x^9 y^{10} v_1^3-15291 x^8 y^{11} v_1^3-10073 x^7 y^{12} v_1^3-5292 x^6 y^{13} v_1^3-2163 x^5 y^{14} v_1^3-660 x^4 y^{15} v_1^3-140 x^3 y^{16} v_1^3-18 x^2 y^{17} v_1^3-x y^{18} v_1^3\right) d^{19}+\left(x^{24} y v_1^4+30 x^{23} y^2 v_1^4+370 x^{22} y^3 v_1^4+2695 x^{21} y^4 v_1^4+13482 x^{20} y^5 v_1^4+50232 x^{19} y^6 v_1^4+146417 x^{18} y^7 v_1^4+344727 x^{17} y^8 v_1^4+669895 x^{16} y^9 v_1^4+1090386 x^{15} y^{10} v_1^4+1501281 x^{14} y^{11} v_1^4+1759044 x^{13} y^{12} v_1^4+1759044 x^{12} y^{13} v_1^4+1501281 x^{11} y^{14} v_1^4+1090386 x^{10} y^{15} v_1^4+669895 x^9 y^{16} v_1^4+344727 x^8 y^{17} v_1^4+146417 x^7 y^{18} v_1^4+50232 x^6 y^{19} v_1^4+13482 x^5 y^{20} v_1^4+2695 x^4 y^{21} v_1^4+370 x^3 y^{22} v_1^4+30 x^2 y^{23} v_1^4+x y^{24} v_1^4\right) d^{25}+\left(-x^{30} y v_1^5-45 x^{29} y^2 v_1^5-805 x^{28} y^3 v_1^5-8330 x^{27} y^4 v_1^5-58464 x^{26} y^5 v_1^5-303576 x^{25} y^6 v_1^5-1230617 x^{24} y^7 v_1^5-4036575 x^{23} y^8 v_1^5-10985510 x^{22} y^9 v_1^5-25257694 x^{21} y^{10} v_1^5-49715370 x^{20} y^{11} v_1^5-84595524 x^{19} y^{12} v_1^5-125325240 x^{18} y^{13} v_1^5-162445482 x^{17} y^{14} v_1^5-184804575 x^{16} y^{15} v_1^5-184804575 x^{15} y^{16} v_1^5-162445482 x^{14} y^{17} v_1^5-125325240 x^{13} y^{18} v_1^5-84595524 x^{12} y^{19} v_1^5-49715370 x^{11} y^{20} v_1^5-25257694 x^{10} y^{21} v_1^5-10985510 x^9 y^{22} v_1^5-4036575 x^8 y^{23} v_1^5-1230617 x^7 y^{24} v_1^5-303576 x^6 y^{25} v_1^5-58464 x^5 y^{26} v_1^5-8330 x^4 y^{27} v_1^5-805 x^3 y^{28} v_1^5-45 x^2 y^{29} v_1^5-x y^{30} v_1^5\right) d^{31}+\left(x^{36} y v_1^6+63 x^{35} y^2 v_1^6+1540 x^{34} y^3 v_1^6+21420 x^{33} y^4 v_1^6+199836 x^{32} y^5 v_1^6+1369368 x^{31} y^6 v_1^6+7294961 x^{30} y^7 v_1^6+31392675 x^{29} y^8 v_1^6+112139540 x^{28} y^9 v_1^6+339246152 x^{27} y^{10} v_1^6+882390024 x^{26} y^{11} v_1^6+1996313904 x^{25} y^{12} v_1^6+3963806640 x^{24} y^{13} v_1^6+6955432950 x^{23} y^{14} v_1^6+10843612350 x^{22} y^{15} v_1^6+15079666480 x^{21} y^{16} v_1^6+18759068100 x^{20} y^{17} v_1^6+20913494940 x^{19} y^{18} v_1^6+20913494940 x^{18} y^{19} v_1^6+18759068100 x^{17} y^{20} v_1^6+15079666480 x^{16} y^{21} v_1^6+10843612350 x^{15} y^{22} v_1^6+6955432950 x^{14} y^{23} v_1^6+3963806640 x^{13} y^{24} v_1^6+1996313904 x^{12} y^{25} v_1^6+882390024 x^{11} y^{26} v_1^6+339246152 x^{10} y^{27} v_1^6+112139540 x^9 y^{28} v_1^6+31392675 x^8 y^{29} v_1^6+7294961 x^7 y^{30} v_1^6+1369368 x^6 y^{31} v_1^6+199836 x^5 y^{32} v_1^6+21420 x^4 y^{33} v_1^6+1540 x^3 y^{34} v_1^6+63 x^2 y^{35} v_1^6+x y^{36} v_1^6\right) d^{37}+\left(-x^{42} y v_1^7-84 x^{41} y^2 v_1^7-2688 x^{40} y^3 v_1^7-48300 x^{39} y^4 v_1^7-576576 x^{38} y^5 v_1^7-5021016 x^{37} y^6 v_1^7-33834617 x^{36} y^7 v_1^7-183648447 x^{35} y^8 v_1^7-826327700 x^{34} y^9 v_1^7-3148755096 x^{33} y^{10} v_1^7-10328591052 x^{32} y^{11} v_1^7-29538690480 x^{31} y^{12} v_1^7-74398956984 x^{30} y^{13} v_1^7-166366137285 x^{29} y^{14} v_1^7-332424118596 x^{28} y^{15} v_1^7-596625629828 x^{27} y^{16} v_1^7-965802148056 x^{26} y^{17} v_1^7-1414686478764 x^{25} y^{18} v_1^7-1879716343440 x^{24} y^{19} v_1^7-2269662112260 x^{23} y^{20} v_1^7-2493282220010 x^{22} y^{21} v_1^7-2493282220010 x^{21} y^{22} v_1^7-2269662112260 x^{20} y^{23} v_1^7-1879716343440 x^{19} y^{24} v_1^7-1414686478764 x^{18} y^{25} v_1^7-965802148056 x^{17} y^{26} v_1^7-596625629828 x^{16} y^{27} v_1^7-332424118596 x^{15} y^{28} v_1^7-166366137285 x^{14} y^{29} v_1^7-74398956984 x^{13} y^{30} v_1^7-29538690480 x^{12} y^{31} v_1^7-10328591052 x^{11} y^{32} v_1^7-3148755096 x^{10} y^{33} v_1^7-826327700 x^9 y^{34} v_1^7-183648447 x^8 y^{35} v_1^7-33834617 x^7 y^{36} v_1^7-5021016 x^6 y^{37} v_1^7-576576 x^5 y^{38} v_1^7-48300 x^4 y^{39} v_1^7-2688 x^3 y^{40} v_1^7-84 x^2 y^{41} v_1^7-x y^{42} v_1^7\right) d^{43}+\left(x^{25} y^{24} \left(307645578441240 v_1^8-9029329031268 v_2\right)+x^{24} y^{25} \left(307645578441240 v_1^8-9029329031268 v_2\right)+x^{26} y^{23} \left(283211213315256 v_1^8-8334765259632 v_2\right)+x^{23} y^{26} \left(283211213315256 v_1^8-8334765259632 v_2\right)+x^{27} y^{22} \left(239917206284232 v_1^8-7099985221168 v_2\right)+x^{22} y^{27} \left(239917206284232 v_1^8-7099985221168 v_2\right)+x^{28} y^{21} \left(186879793026199 v_1^8-5578559816632 v_2\right)+x^{21} y^{28} \left(186879793026199 v_1^8-5578559816632 v_2\right)+x^{29} y^{20} \left(133687632583146 v_1^8-4039646763768 v_2\right)+x^{20} y^{29} \left(133687632583146 v_1^8-4039646763768 v_2\right)+x^{30} y^{19} \left(87686379270908 v_1^8-2693097842512 v_2\right)+x^{19} y^{30} \left(87686379270908 v_1^8-2693097842512 v_2\right)+x^{31} y^{18} \left(52620719194412 v_1^8-1650608355088 v_2\right)+x^{18} y^{31} \left(52620719194412 v_1^8-1650608355088 v_2\right)+x^{32} y^{17} \left(28813721993604 v_1^8-928467199737 v_2\right)+x^{17} y^{32} \left(28813721993604 v_1^8-928467199737 v_2\right)+x^{33} y^{16} \left(14349046857348 v_1^8-478301284713 v_2\right)+x^{16} y^{33} \left(14349046857348 v_1^8-478301284713 v_2\right)+x^{34} y^{15} \left(6472553964180 v_1^8-225082957512 v_2\right)+x^{15} y^{34} \left(6472553964180 v_1^8-225082957512 v_2\right)+x^{35} y^{14} \left(2631667867983 v_1^8-96464124648 v_2\right)+x^{14} y^{35} \left(2631667867983 v_1^8-96464124648 v_2\right)+x^{36} y^{13} \left(958762285444 v_1^8-37513826252 v_2\right)+x^{13} y^{36} \left(958762285444 v_1^8-37513826252 v_2\right)+x^{37} y^{12} \left(310727271556 v_1^8-13180533548 v_2\right)+x^{12} y^{37} \left(310727271556 v_1^8-13180533548 v_2\right)+x^{38} y^{11} \left(88796970600 v_1^8-4162273752 v_2\right)+x^{11} y^{38} \left(88796970600 v_1^8-4162273752 v_2\right)+x^{39} y^{10} \left(22132155352 v_1^8-1173974648 v_2\right)+x^{10} y^{39} \left(22132155352 v_1^8-1173974648 v_2\right)+x^{40} y^9 \left(4745852752 v_1^8-293493662 v_2\right)+x^9 y^{40} \left(4745852752 v_1^8-293493662 v_2\right)+x^{41} y^8 \left(860383632 v_1^8-64425438 v_2\right)+x^8 y^{41} \left(860383632 v_1^8-64425438 v_2\right)+x^{42} y^7 \left(128901941 v_1^8-12271512 v_2\right)+x^7 y^{42} \left(128901941 v_1^8-12271512 v_2\right)+x^{43} y^6 \left(15476076 v_1^8-1997688 v_2\right)+x^6 y^{43} \left(15476076 v_1^8-1997688 v_2\right)+x^{44} y^5 \left(1425690 v_1^8-272412 v_2\right)+x^5 y^{44} \left(1425690 v_1^8-272412 v_2\right)+x^{45} y^4 \left(94346 v_1^8-30268 v_2\right)+x^4 y^{45} \left(94346 v_1^8-30268 v_2\right)+x^{46} y^3 \left(4004 v_1^8-2632 v_2\right)+x^3 y^{46} \left(4004 v_1^8-2632 v_2\right)+x^{47} y^2 \left(84 v_1^8-168 v_2\right)+x^2 y^{47} \left(84 v_1^8-168 v_2\right)-7 x^{48} y v_2-7 x y^{48} v_2\right) d^{49}+\left(x^{54} y \left(v_1^9+14 v_1 v_2\right)+x y^{54} \left(v_1^9+14 v_1 v_2\right)+x^{53} y^2 \left(-54 v_1^9+567 v_1 v_2\right)+x^2 y^{53} \left(-54 v_1^9+567 v_1 v_2\right)+x^{52} y^3 \left(-4953 v_1^9+12684 v_1 v_2\right)+x^3 y^{52} \left(-4953 v_1^9+12684 v_1 v_2\right)+x^{51} y^4 \left(-158730 v_1^9+195195 v_1 v_2\right)+x^4 y^{51} \left(-158730 v_1^9+195195 v_1 v_2\right)+x^{50} y^5 \left(-3044733 v_1^9+2263422 v_1 v_2\right)+x^5 y^{50} \left(-3044733 v_1^9+2263422 v_1 v_2\right)+x^{49} y^6 \left(-40848850 v_1^9+20859545 v_1 v_2\right)+x^6 y^{49} \left(-40848850 v_1^9+20859545 v_1 v_2\right)+x^{48} y^7 \left(-414843891 v_1^9+158288327 v_1 v_2\right)+x^7 y^{48} \left(-414843891 v_1^9+158288327 v_1 v_2\right)+x^{47} y^8 \left(-3349446978 v_1^9+1014155358 v_1 v_2\right)+x^8 y^{47} \left(-3349446978 v_1^9+1014155358 v_1 v_2\right)+x^{46} y^9 \left(-22237408754 v_1^9+5589637432 v_1 v_2\right)+x^9 y^{46} \left(-22237408754 v_1^9+5589637432 v_1 v_2\right)+x^{45} y^{10} \left(-124424217202 v_1^9+26886294728 v_1 v_2\right)+x^{10} y^{45} \left(-124424217202 v_1^9+26886294728 v_1 v_2\right)+x^{44} y^{11} \left(-597804745920 v_1^9+114151537452 v_1 v_2\right)+x^{11} y^{44} \left(-597804745920 v_1^9+114151537452 v_1 v_2\right)+x^{43} y^{12} \left(-2502672779066 v_1^9+431735172028 v_1 v_2\right)+x^{12} y^{43} \left(-2502672779066 v_1^9+431735172028 v_1 v_2\right)+x^{42} y^{13} \left(-9236782595334 v_1^9+1465554325992 v_1 v_2\right)+x^{13} y^{42} \left(-9236782595334 v_1^9+1465554325992 v_1 v_2\right)+x^{41} y^{14} \left(-30341628948162 v_1^9+4493090288088 v_1 v_2\right)+x^{14} y^{41} \left(-30341628948162 v_1^9+4493090288088 v_1 v_2\right)+x^{40} y^{15} \left(-89403988040562 v_1^9+12506020315422 v_1 v_2\right)+x^{15} y^{40} \left(-89403988040562 v_1^9+12506020315422 v_1 v_2\right)+x^{39} y^{16} \left(-237847152326873 v_1^9+31742618339113 v_1 v_2\right)+x^{16} y^{39} \left(-237847152326873 v_1^9+31742618339113 v_1 v_2\right)+x^{38} y^{17} \left(-574412297681211 v_1^9+73747074859392 v_1 v_2\right)+x^{17} y^{38} \left(-574412297681211 v_1^9+73747074859392 v_1 v_2\right)+x^{37} y^{18} \left(-1265081442916813 v_1^9+157330090480328 v_1 v_2\right)+x^{18} y^{37} \left(-1265081442916813 v_1^9+157330090480328 v_1 v_2\right)+x^{36} y^{19} \left(-2550660930790619 v_1^9+309047080370452 v_1 v_2\right)+x^{19} y^{36} \left(-2550660930790619 v_1^9+309047080370452 v_1 v_2\right)+x^{35} y^{20} \left(-4723151535892461 v_1^9+560256866543328 v_1 v_2\right)+x^{20} y^{35} \left(-4723151535892461 v_1^9+560256866543328 v_1 v_2\right)+x^{34} y^{21} \left(-8054412906400329 v_1^9+939179230514432 v_1 v_2\right)+x^{21} y^{34} \left(-8054412906400329 v_1^9+939179230514432 v_1 v_2\right)+x^{33} y^{22} \left(-12677643205503735 v_1^9+1458210940536408 v_1 v_2\right)+x^{22} y^{33} \left(-12677643205503735 v_1^9+1458210940536408 v_1 v_2\right)+x^{32} y^{23} \left(-18452285440938333 v_1^9+2099864204185977 v_1 v_2\right)+x^{23} y^{32} \left(-18452285440938333 v_1^9+2099864204185977 v_1 v_2\right)+x^{31} y^{24} \left(-24872274537034212 v_1^9+2807610311679588 v_1 v_2\right)+x^{24} y^{31} \left(-24872274537034212 v_1^9+2807610311679588 v_1 v_2\right)+x^{30} y^{25} \left(-31084016312562592 v_1^9+3488419361153648 v_1 v_2\right)+x^{25} y^{30} \left(-31084016312562592 v_1^9+3488419361153648 v_1 v_2\right)+x^{29} y^{26} \left(-36048207290190276 v_1^9+4030326607541712 v_1 v_2\right)+x^{26} y^{29} \left(-36048207290190276 v_1^9+4030326607541712 v_1 v_2\right)+x^{28} y^{27} \left(-38814771653343742 v_1^9+4331630424575256 v_1 v_2\right)+x^{27} y^{28} \left(-38814771653343742 v_1^9+4331630424575256 v_1 v_2\right)\right) d^{55}+O[d]^{56}$
$(x+y) d+\left(-x^{10} y v_1-5 x^9 y^2 v_1-15 x^8 y^3 v_1-30 x^7 y^4 v_1-42 x^6 y^5 v_1-42 x^5 y^6 v_1-30 x^4 y^7 v_1-15 x^3 y^8 v_1-5 x^2 y^9 v_1-x y^{10} v_1\right) d^{11}+\left(x^{20} y v_1^2+15 x^{19} y^2 v_1^2+110 x^{18} y^3 v_1^2+525 x^{17} y^4 v_1^2+1827 x^{16} y^5 v_1^2+4914 x^{15} y^6 v_1^2+10560 x^{14} y^7 v_1^2+18495 x^{13} y^8 v_1^2+26720 x^{12} y^9 v_1^2+32065 x^{11} y^{10} v_1^2+32065 x^{10} y^{11} v_1^2+26720 x^9 y^{12} v_1^2+18495 x^8 y^{13} v_1^2+10560 x^7 y^{14} v_1^2+4914 x^6 y^{15} v_1^2+1827 x^5 y^{16} v_1^2+525 x^4 y^{17} v_1^2+110 x^3 y^{18} v_1^2+15 x^2 y^{19} v_1^2+x y^{20} v_1^2\right) d^{21}+\left(-x^{30} y v_1^3-30 x^{29} y^2 v_1^3-400 x^{28} y^3 v_1^3-3325 x^{27} y^4 v_1^3-19782 x^{26} y^5 v_1^3-90636 x^{25} y^6 v_1^3-334260 x^{24} y^7 v_1^3-1021275 x^{23} y^8 v_1^3-2636645 x^{22} y^9 v_1^3-5832684 x^{21} y^{10} v_1^3-11167189 x^{20} y^{11} v_1^3-18638700 x^{19} y^{12} v_1^3-27259650 x^{18} y^{13} v_1^3-35058540 x^{17} y^{14} v_1^3-39737331 x^{16} y^{15} v_1^3-39737331 x^{15} y^{16} v_1^3-35058540 x^{14} y^{17} v_1^3-27259650 x^{13} y^{18} v_1^3-18638700 x^{12} y^{19} v_1^3-11167189 x^{11} y^{20} v_1^3-5832684 x^{10} y^{21} v_1^3-2636645 x^9 y^{22} v_1^3-1021275 x^8 y^{23} v_1^3-334260 x^7 y^{24} v_1^3-90636 x^6 y^{25} v_1^3-19782 x^5 y^{26} v_1^3-3325 x^4 y^{27} v_1^3-400 x^3 y^{28} v_1^3-30 x^2 y^{29} v_1^3-x y^{30} v_1^3\right) d^{31}+\left(x^{40} y v_1^4+50 x^{39} y^2 v_1^4+1050 x^{38} y^3 v_1^4+13300 x^{37} y^4 v_1^4+118202 x^{36} y^5 v_1^4+799848 x^{35} y^6 v_1^4+4333500 x^{34} y^7 v_1^4+19438650 x^{33} y^8 v_1^4+73911695 x^{32} y^9 v_1^4+242350108 x^{31} y^{10} v_1^4+694153857 x^{30} y^{11} v_1^4+1754023340 x^{29} y^{12} v_1^4+3940080880 x^{28} y^{13} v_1^4+7915219500 x^{27} y^{14} v_1^4+14287126446 x^{26} y^{15} v_1^4+23256285660 x^{25} y^{16} v_1^4+34235345340 x^{24} y^{17} v_1^4+45673941090 x^{23} y^{18} v_1^4+55306910055 x^{22} y^{19} v_1^4+60845867940 x^{21} y^{20} v_1^4+60845867940 x^{20} y^{21} v_1^4+55306910055 x^{19} y^{22} v_1^4+45673941090 x^{18} y^{23} v_1^4+34235345340 x^{17} y^{24} v_1^4+23256285660 x^{16} y^{25} v_1^4+14287126446 x^{15} y^{26} v_1^4+7915219500 x^{14} y^{27} v_1^4+3940080880 x^{13} y^{28} v_1^4+1754023340 x^{12} y^{29} v_1^4+694153857 x^{11} y^{30} v_1^4+242350108 x^{10} y^{31} v_1^4+73911695 x^9 y^{32} v_1^4+19438650 x^8 y^{33} v_1^4+4333500 x^7 y^{34} v_1^4+799848 x^6 y^{35} v_1^4+118202 x^5 y^{36} v_1^4+13300 x^4 y^{37} v_1^4+1050 x^3 y^{38} v_1^4+50 x^2 y^{39} v_1^4+x y^{40} v_1^4\right) d^{41}+\left(-x^{50} y v_1^5-75 x^{49} y^2 v_1^5-2275 x^{48} y^3 v_1^5-40600 x^{47} y^4 v_1^5-499842 x^{46} y^5 v_1^5-4631970 x^{45} y^6 v_1^5-34110450 x^{44} y^7 v_1^5-207046125 x^{43} y^8 v_1^5-1063132070 x^{42} y^9 v_1^5-4707504802 x^{41} y^{10} v_1^5-18240308119 x^{40} y^{11} v_1^5-62555050400 x^{39} y^{12} v_1^5-191605231930 x^{38} y^{13} v_1^5-527986560460 x^{37} y^{14} v_1^5-1316653942774 x^{36} y^{15} v_1^5-2985727390947 x^{35} y^{16} v_1^5-6181319503485 x^{34} y^{17} v_1^5-11721491484395 x^{33} y^{18} v_1^5-20413653094770 x^{32} y^{19} v_1^5-32722572560860 x^{31} y^{20} v_1^5-48365238083812 x^{30} y^{21} v_1^5-66006957723630 x^{29} y^{22} v_1^5-83269626432760 x^{28} y^{23} v_1^5-97177536089200 x^{27} y^{24} v_1^5-104966446824936 x^{26} y^{25} v_1^5-104966446824936 x^{25} y^{26} v_1^5-97177536089200 x^{24} y^{27} v_1^5-83269626432760 x^{23} y^{28} v_1^5-66006957723630 x^{22} y^{29} v_1^5-48365238083812 x^{21} y^{30} v_1^5-32722572560860 x^{20} y^{31} v_1^5-20413653094770 x^{19} y^{32} v_1^5-11721491484395 x^{18} y^{33} v_1^5-6181319503485 x^{17} y^{34} v_1^5-2985727390947 x^{16} y^{35} v_1^5-1316653942774 x^{15} y^{36} v_1^5-527986560460 x^{14} y^{37} v_1^5-191605231930 x^{13} y^{38} v_1^5-62555050400 x^{12} y^{39} v_1^5-18240308119 x^{11} y^{40} v_1^5-4707504802 x^{10} y^{41} v_1^5-1063132070 x^9 y^{42} v_1^5-207046125 x^8 y^{43} v_1^5-34110450 x^7 y^{44} v_1^5-4631970 x^6 y^{45} v_1^5-499842 x^5 y^{46} v_1^5-40600 x^4 y^{47} v_1^5-2275 x^3 y^{48} v_1^5-75 x^2 y^{49} v_1^5-x y^{50} v_1^5\right) d^{51}+\left(x^{60} y v_1^6+105 x^{59} y^2 v_1^6+4340 x^{58} y^3 v_1^6+103530 x^{57} y^4 v_1^6+1680084 x^{56} y^5 v_1^6+20312754 x^{55} y^6 v_1^6+193710660 x^{54} y^7 v_1^6+1514593080 x^{53} y^8 v_1^6+9982402430 x^{52} y^9 v_1^6+56615997438 x^{51} y^{10} v_1^6+280732659877 x^{50} y^{11} v_1^6+1232274466550 x^{49} y^{12} v_1^6+4836332067105 x^{48} y^{13} v_1^6+17109696497020 x^{47} y^{14} v_1^6+54927036172890 x^{46} y^{15} v_1^6+160900954950960 x^{45} y^{16} v_1^6+432095599759635 x^{44} y^{17} v_1^6+1067955096404625 x^{43} y^{18} v_1^6+2437364192169270 x^{42} y^{19} v_1^6+5151185143538980 x^{41} y^{20} v_1^6+10105431803674350 x^{40} y^{21} v_1^6+18439486163844050 x^{39} y^{22} v_1^6+31350118354387210 x^{38} y^{23} v_1^6+49734561555585060 x^{37} y^{24} v_1^6+73711336128981344 x^{36} y^{25} v_1^6+102164993412262956 x^{35} y^{26} v_1^6+132529409720182545 x^{34} y^{27} v_1^6+161004332684400190 x^{33} y^{28} v_1^6+183264495556558845 x^{32} y^{29} v_1^6+195508719268445492 x^{31} y^{30} v_1^6+195508719268445492 x^{30} y^{31} v_1^6+183264495556558845 x^{29} y^{32} v_1^6+161004332684400190 x^{28} y^{33} v_1^6+132529409720182545 x^{27} y^{34} v_1^6+102164993412262956 x^{26} y^{35} v_1^6+73711336128981344 x^{25} y^{36} v_1^6+49734561555585060 x^{24} y^{37} v_1^6+31350118354387210 x^{23} y^{38} v_1^6+18439486163844050 x^{22} y^{39} v_1^6+10105431803674350 x^{21} y^{40} v_1^6+5151185143538980 x^{20} y^{41} v_1^6+2437364192169270 x^{19} y^{42} v_1^6+1067955096404625 x^{18} y^{43} v_1^6+432095599759635 x^{17} y^{44} v_1^6+160900954950960 x^{16} y^{45} v_1^6+54927036172890 x^{15} y^{46} v_1^6+17109696497020 x^{14} y^{47} v_1^6+4836332067105 x^{13} y^{48} v_1^6+1232274466550 x^{12} y^{49} v_1^6+280732659877 x^{11} y^{50} v_1^6+56615997438 x^{10} y^{51} v_1^6+9982402430 x^9 y^{52} v_1^6+1514593080 x^8 y^{53} v_1^6+193710660 x^7 y^{54} v_1^6+20312754 x^6 y^{55} v_1^6+1680084 x^5 y^{56} v_1^6+103530 x^4 y^{57} v_1^6+4340 x^3 y^{58} v_1^6+105 x^2 y^{59} v_1^6+x y^{60} v_1^6\right) d^{61}+\left(-x^{70} y v_1^7-140 x^{69} y^2 v_1^7-7560 x^{68} y^3 v_1^7-232050 x^{67} y^4 v_1^7-4789554 x^{66} y^5 v_1^7-72997848 x^{65} y^6 v_1^7-871547820 x^{64} y^7 v_1^7-8486975640 x^{63} y^8 v_1^7-69391231910 x^{62} y^9 v_1^7-486841635280 x^{61} y^{10} v_1^7-2980490819157 x^{60} y^{11} v_1^7-16134728562330 x^{59} y^{12} v_1^7-78063177080280 x^{58} y^{13} v_1^7-340514287240200 x^{57} y^{14} v_1^7-1348881327292236 x^{56} y^{15} v_1^7-4881985594593492 x^{55} y^{16} v_1^7-16226754810667905 x^{54} y^{17} v_1^7-49748218947276360 x^{53} y^{18} v_1^7-141208707557022630 x^{52} y^{19} v_1^7-372293798837551500 x^{51} y^{20} v_1^7-914247377198877072 x^{50} y^{21} v_1^7-2096273796314337675 x^{49} y^{22} v_1^7-4497322102613243085 x^{48} y^{23} v_1^7-9044369094117937020 x^{47} y^{24} v_1^7-17077093066841288544 x^{46} y^{25} v_1^7-30315386317682852190 x^{45} y^{26} v_1^7-50657905104266684595 x^{44} y^{27} v_1^7-79765604774875282270 x^{43} y^{28} v_1^7-118454819090401101560 x^{42} y^{29} v_1^7-166028843135960950698 x^{41} y^{30} v_1^7-219775230331639860990 x^{40} y^{31} v_1^7-274889670620351792145 x^{39} y^{32} v_1^7-325008822945815611575 x^{38} y^{33} v_1^7-363342646099235668600 x^{37} y^{34} v_1^7-384154671476102636984 x^{36} y^{35} v_1^7-384154671476102636984 x^{35} y^{36} v_1^7-363342646099235668600 x^{34} y^{37} v_1^7-325008822945815611575 x^{33} y^{38} v_1^7-274889670620351792145 x^{32} y^{39} v_1^7-219775230331639860990 x^{31} y^{40} v_1^7-166028843135960950698 x^{30} y^{41} v_1^7-118454819090401101560 x^{29} y^{42} v_1^7-79765604774875282270 x^{28} y^{43} v_1^7-50657905104266684595 x^{27} y^{44} v_1^7-30315386317682852190 x^{26} y^{45} v_1^7-17077093066841288544 x^{25} y^{46} v_1^7-9044369094117937020 x^{24} y^{47} v_1^7-4497322102613243085 x^{23} y^{48} v_1^7-2096273796314337675 x^{22} y^{49} v_1^7-914247377198877072 x^{21} y^{50} v_1^7-372293798837551500 x^{20} y^{51} v_1^7-141208707557022630 x^{19} y^{52} v_1^7-49748218947276360 x^{18} y^{53} v_1^7-16226754810667905 x^{17} y^{54} v_1^7-4881985594593492 x^{16} y^{55} v_1^7-1348881327292236 x^{15} y^{56} v_1^7-340514287240200 x^{14} y^{57} v_1^7-78063177080280 x^{13} y^{58} v_1^7-16134728562330 x^{12} y^{59} v_1^7-2980490819157 x^{11} y^{60} v_1^7-486841635280 x^{10} y^{61} v_1^7-69391231910 x^9 y^{62} v_1^7-8486975640 x^8 y^{63} v_1^7-871547820 x^7 y^{64} v_1^7-72997848 x^6 y^{65} v_1^7-4789554 x^5 y^{66} v_1^7-232050 x^4 y^{67} v_1^7-7560 x^3 y^{68} v_1^7-140 x^2 y^{69} v_1^7-x y^{70} v_1^7\right) d^{71}+O[d]^{72}$
$(x+y) d+\left(-x^{12} y v_1-6 x^{11} y^2 v_1-22 x^{10} y^3 v_1-55 x^9 y^4 v_1-99 x^8 y^5 v_1-132 x^7 y^6 v_1-132 x^6 y^7 v_1-99 x^5 y^8 v_1-55 x^4 y^9 v_1-22 x^3 y^{10} v_1-6 x^2 y^{11} v_1-x y^{12} v_1\right) d^{13}+\left(x^{24} y v_1^2+18 x^{23} y^2 v_1^2+160 x^{22} y^3 v_1^2+935 x^{21} y^4 v_1^2+4026 x^{20} y^5 v_1^2+13552 x^{19} y^6 v_1^2+36916 x^{18} y^7 v_1^2+83160 x^{17} y^8 v_1^2+157135 x^{16} y^9 v_1^2+251438 x^{15} y^{10} v_1^2+342876 x^{14} y^{11} v_1^2+400023 x^{13} y^{12} v_1^2+400023 x^{12} y^{13} v_1^2+342876 x^{11} y^{14} v_1^2+251438 x^{10} y^{15} v_1^2+157135 x^9 y^{16} v_1^2+83160 x^8 y^{17} v_1^2+36916 x^7 y^{18} v_1^2+13552 x^6 y^{19} v_1^2+4026 x^5 y^{20} v_1^2+935 x^4 y^{21} v_1^2+160 x^3 y^{22} v_1^2+18 x^2 y^{23} v_1^2+x y^{24} v_1^2\right) d^{25}+\left(-x^{36} y v_1^3-36 x^{35} y^2 v_1^3-580 x^{34} y^3 v_1^3-5865 x^{33} y^4 v_1^3-42735 x^{32} y^5 v_1^3-241472 x^{31} y^6 v_1^3-1106292 x^{30} y^7 v_1^3-4231755 x^{29} y^8 v_1^3-13792790 x^{28} y^9 v_1^3-38871250 x^{27} y^{10} v_1^3-95754126 x^{26} y^{11} v_1^3-207867296 x^{25} y^{12} v_1^3-400144823 x^{24} y^{13} v_1^3-686305428 x^{23} y^{14} v_1^3-1052586400 x^{22} y^{15} v_1^3-1447463215 x^{21} y^{16} v_1^3-1788124800 x^{20} y^{17} v_1^3-1986837776 x^{19} y^{18} v_1^3-1986837776 x^{18} y^{19} v_1^3-1788124800 x^{17} y^{20} v_1^3-1447463215 x^{16} y^{21} v_1^3-1052586400 x^{15} y^{22} v_1^3-686305428 x^{14} y^{23} v_1^3-400144823 x^{13} y^{24} v_1^3-207867296 x^{12} y^{25} v_1^3-95754126 x^{11} y^{26} v_1^3-38871250 x^{10} y^{27} v_1^3-13792790 x^9 y^{28} v_1^3-4231755 x^8 y^{29} v_1^3-1106292 x^7 y^{30} v_1^3-241472 x^6 y^{31} v_1^3-42735 x^5 y^{32} v_1^3-5865 x^4 y^{33} v_1^3-580 x^3 y^{34} v_1^3-36 x^2 y^{35} v_1^3-x y^{36} v_1^3\right) d^{37}+\left(x^{48} y v_1^4+60 x^{47} y^2 v_1^4+1520 x^{46} y^3 v_1^4+23345 x^{45} y^4 v_1^4+252840 x^{44} y^5 v_1^4+2095632 x^{43} y^6 v_1^4+13979460 x^{42} y^7 v_1^4+77623920 x^{41} y^8 v_1^4+367412870 x^{40} y^9 v_1^4+1508522730 x^{39} y^{10} v_1^4+5444152896 x^{38} y^{11} v_1^4+17447684800 x^{37} y^{12} v_1^4+50058940023 x^{36} y^{13} v_1^4+129409294056 x^{35} y^{14} v_1^4+303007605780 x^{34} y^{15} v_1^4+645338624265 x^{33} y^{16} v_1^4+1254504266400 x^{32} y^{17} v_1^4+2232216568736 x^{31} y^{18} v_1^4+3644024003312 x^{30} y^{19} v_1^4+5467822470330 x^{29} y^{20} v_1^4+7552244078390 x^{28} y^{21} v_1^4+9612982040800 x^{27} y^{22} v_1^4+11285445678378 x^{26} y^{23} v_1^4+12226195896096 x^{25} y^{24} v_1^4+12226195896096 x^{24} y^{25} v_1^4+11285445678378 x^{23} y^{26} v_1^4+9612982040800 x^{22} y^{27} v_1^4+7552244078390 x^{21} y^{28} v_1^4+5467822470330 x^{20} y^{29} v_1^4+3644024003312 x^{19} y^{30} v_1^4+2232216568736 x^{18} y^{31} v_1^4+1254504266400 x^{17} y^{32} v_1^4+645338624265 x^{16} y^{33} v_1^4+303007605780 x^{15} y^{34} v_1^4+129409294056 x^{14} y^{35} v_1^4+50058940023 x^{13} y^{36} v_1^4+17447684800 x^{12} y^{37} v_1^4+5444152896 x^{11} y^{38} v_1^4+1508522730 x^{10} y^{39} v_1^4+367412870 x^9 y^{40} v_1^4+77623920 x^8 y^{41} v_1^4+13979460 x^7 y^{42} v_1^4+2095632 x^6 y^{43} v_1^4+252840 x^5 y^{44} v_1^4+23345 x^4 y^{45} v_1^4+1520 x^3 y^{46} v_1^4+60 x^2 y^{47} v_1^4+x y^{48} v_1^4\right) d^{49}+\left(-x^{60} y v_1^5-90 x^{59} y^2 v_1^5-3290 x^{58} y^3 v_1^5-71050 x^{57} y^4 v_1^5-1062810 x^{56} y^5 v_1^5-12015192 x^{55} y^6 v_1^5-108384540 x^{54} y^7 v_1^5-809219565 x^{53} y^8 v_1^5-5132816975 x^{52} y^9 v_1^5-28199171000 x^{51} y^{10} v_1^5-136185763896 x^{50} y^{11} v_1^5-584888367700 x^{49} y^{12} v_1^5-2254638172123 x^{48} y^{13} v_1^5-7859597312760 x^{47} y^{14} v_1^5-24929745852240 x^{46} y^{15} v_1^5-72318357945085 x^{45} y^{16} v_1^5-192685451706300 x^{44} y^{17} v_1^5-473241097899416 x^{43} y^{18} v_1^5-1074663346085560 x^{42} y^{19} v_1^5-2262260819893140 x^{41} y^{20} v_1^5-4424347026604010 x^{40} y^{21} v_1^5-8053879635115600 x^{39} y^{22} v_1^5-13667861399466288 x^{38} y^{23} v_1^5-21652998125142300 x^{37} y^{24} v_1^5-32058637598533196 x^{36} y^{25} v_1^5-44400098962038156 x^{35} y^{26} v_1^5-57565129068931300 x^{34} y^{27} v_1^5-69907698175687950 x^{33} y^{28} v_1^5-79554872774991075 x^{32} y^{29} v_1^5-84860836846109632 x^{31} y^{30} v_1^5-84860836846109632 x^{30} y^{31} v_1^5-79554872774991075 x^{29} y^{32} v_1^5-69907698175687950 x^{28} y^{33} v_1^5-57565129068931300 x^{27} y^{34} v_1^5-44400098962038156 x^{26} y^{35} v_1^5-32058637598533196 x^{25} y^{36} v_1^5-21652998125142300 x^{24} y^{37} v_1^5-13667861399466288 x^{23} y^{38} v_1^5-8053879635115600 x^{22} y^{39} v_1^5-4424347026604010 x^{21} y^{40} v_1^5-2262260819893140 x^{20} y^{41} v_1^5-1074663346085560 x^{19} y^{42} v_1^5-473241097899416 x^{18} y^{43} v_1^5-192685451706300 x^{17} y^{44} v_1^5-72318357945085 x^{16} y^{45} v_1^5-24929745852240 x^{15} y^{46} v_1^5-7859597312760 x^{14} y^{47} v_1^5-2254638172123 x^{13} y^{48} v_1^5-584888367700 x^{12} y^{49} v_1^5-136185763896 x^{11} y^{50} v_1^5-28199171000 x^{10} y^{51} v_1^5-5132816975 x^9 y^{52} v_1^5-809219565 x^8 y^{53} v_1^5-108384540 x^7 y^{54} v_1^5-12015192 x^6 y^{55} v_1^5-1062810 x^5 y^{56} v_1^5-71050 x^4 y^{57} v_1^5-3290 x^3 y^{58} v_1^5-90 x^2 y^{59} v_1^5-x y^{60} v_1^5\right) d^{61}+\left(x^{72} y v_1^6+126 x^{71} y^2 v_1^6+6272 x^{70} y^3 v_1^6+180810 x^{69} y^4 v_1^6+3557988 x^{68} y^5 v_1^6+52339056 x^{67} y^6 v_1^6+609344076 x^{66} y^7 v_1^6+5836308192 x^{65} y^8 v_1^6+47283931695 x^{64} y^9 v_1^6+330816333848 x^{63} y^{10} v_1^6+2030861130480 x^{62} y^{11} v_1^6+11077670875180 x^{61} y^{12} v_1^6+54234478432583 x^{60} y^{13} v_1^6+240293076309540 x^{59} y^{14} v_1^6+970082512669410 x^{58} y^{15} v_1^6+3588867466359770 x^{57} y^{16} v_1^6+12225946956321420 x^{56} y^{17} v_1^6+38509520514259536 x^{55} y^{18} v_1^6+112549591115740240 x^{54} y^{19} v_1^6+306146156539753530 x^{53} y^{20} v_1^6+777078930536999255 x^{52} y^{21} v_1^6+1844785885135910080 x^{51} y^{22} v_1^6+4104279978954844248 x^{50} y^{23} v_1^6+8572236003894059700 x^{49} y^{24} v_1^6+16833640058849689516 x^{48} y^{25} v_1^6+31121885275967916420 x^{47} y^{26} v_1^6+54232685076084426560 x^{46} y^{27} v_1^6+89166420795731917190 x^{45} y^{28} v_1^6+138441130096286327100 x^{44} y^{29} v_1^6+203131569039403553472 x^{43} y^{30} v_1^6+281847348557076533000 x^{42} y^{31} v_1^6+370002789358296203340 x^{41} y^{32} v_1^6+459767532273828073350 x^{40} y^{33} v_1^6+540955339160600071700 x^{39} y^{34} v_1^6+602814232269180132096 x^{38} y^{35} v_1^6+636321543290255902660 x^{37} y^{36} v_1^6+636321543290255902660 x^{36} y^{37} v_1^6+602814232269180132096 x^{35} y^{38} v_1^6+540955339160600071700 x^{34} y^{39} v_1^6+459767532273828073350 x^{33} y^{40} v_1^6+370002789358296203340 x^{32} y^{41} v_1^6+281847348557076533000 x^{31} y^{42} v_1^6+203131569039403553472 x^{30} y^{43} v_1^6+138441130096286327100 x^{29} y^{44} v_1^6+89166420795731917190 x^{28} y^{45} v_1^6+54232685076084426560 x^{27} y^{46} v_1^6+31121885275967916420 x^{26} y^{47} v_1^6+16833640058849689516 x^{25} y^{48} v_1^6+8572236003894059700 x^{24} y^{49} v_1^6+4104279978954844248 x^{23} y^{50} v_1^6+1844785885135910080 x^{22} y^{51} v_1^6+777078930536999255 x^{21} y^{52} v_1^6+306146156539753530 x^{20} y^{53} v_1^6+112549591115740240 x^{19} y^{54} v_1^6+38509520514259536 x^{18} y^{55} v_1^6+12225946956321420 x^{17} y^{56} v_1^6+3588867466359770 x^{16} y^{57} v_1^6+970082512669410 x^{15} y^{58} v_1^6+240293076309540 x^{14} y^{59} v_1^6+54234478432583 x^{13} y^{60} v_1^6+11077670875180 x^{12} y^{61} v_1^6+2030861130480 x^{11} y^{62} v_1^6+330816333848 x^{10} y^{63} v_1^6+47283931695 x^9 y^{64} v_1^6+5836308192 x^8 y^{65} v_1^6+609344076 x^7 y^{66} v_1^6+52339056 x^6 y^{67} v_1^6+3557988 x^5 y^{68} v_1^6+180810 x^4 y^{69} v_1^6+6272 x^3 y^{70} v_1^6+126 x^2 y^{71} v_1^6+x y^{72} v_1^6\right) d^{73}+\left(-x^{84} y v_1^7-168 x^{83} y^2 v_1^7-10920 x^{82} y^3 v_1^7-404670 x^{81} y^4 v_1^7-10113642 x^{80} y^5 v_1^7-187187616 x^{79} y^6 v_1^7-2721890028 x^{78} y^7 v_1^7-32374735965 x^{77} y^8 v_1^7-324267783840 x^{76} y^9 v_1^7-2795251491032 x^{75} y^{10} v_1^7-21089394023880 x^{74} y^{11} v_1^7-141128934022440 x^{73} y^{12} v_1^7-846727723327823 x^{72} y^{13} v_1^7-4594892796281196 x^{71} y^{14} v_1^7-22719241748399808 x^{70} y^{15} v_1^7-102985550115581490 x^{69} y^{16} v_1^7-430226120954124180 x^{68} y^{17} v_1^7-1663808199660842928 x^{67} y^{18} v_1^7-5979662716050990736 x^{66} y^{19} v_1^7-20039033117497187508 x^{65} y^{20} v_1^7-62802657609487339995 x^{64} y^{21} v_1^7-184543426066091279680 x^{63} y^{22} v_1^7-509592793949490565440 x^{62} y^{23} v_1^7-1325020281793686766680 x^{61} y^{24} v_1^7-3249883100605928464776 x^{60} y^{25} v_1^7-7530851992287006452250 x^{59} y^{26} v_1^7-16510538365359116960630 x^{58} y^{27} v_1^7-34289565311011555091860 x^{57} y^{28} v_1^7-67535165894310529697070 x^{56} y^{29} v_1^7-126268751749984740521352 x^{55} y^{30} v_1^7-224306983742929090868480 x^{54} y^{31} v_1^7-378887847928116129232245 x^{53} y^{32} v_1^7-608976122515421047539165 x^{52} y^{33} v_1^7-931915013065557851485680 x^{51} y^{34} v_1^7-1358533430868363708542256 x^{50} y^{35} v_1^7-1887482608471602413816560 x^{49} y^{36} v_1^7-2500264099260704511796880 x^{48} y^{37} v_1^7-3158809886595190016478240 x^{47} y^{38} v_1^7-3807274338451108299620560 x^{46} y^{39} v_1^7-4378762889163210875546450 x^{45} y^{40} v_1^7-4806231405799326136772040 x^{44} y^{41} v_1^7-5035236381763940729380080 x^{43} y^{42} v_1^7-5035236381763940729380080 x^{42} y^{43} v_1^7-4806231405799326136772040 x^{41} y^{44} v_1^7-4378762889163210875546450 x^{40} y^{45} v_1^7-3807274338451108299620560 x^{39} y^{46} v_1^7-3158809886595190016478240 x^{38} y^{47} v_1^7-2500264099260704511796880 x^{37} y^{48} v_1^7-1887482608471602413816560 x^{36} y^{49} v_1^7-1358533430868363708542256 x^{35} y^{50} v_1^7-931915013065557851485680 x^{34} y^{51} v_1^7-608976122515421047539165 x^{33} y^{52} v_1^7-378887847928116129232245 x^{32} y^{53} v_1^7-224306983742929090868480 x^{31} y^{54} v_1^7-126268751749984740521352 x^{30} y^{55} v_1^7-67535165894310529697070 x^{29} y^{56} v_1^7-34289565311011555091860 x^{28} y^{57} v_1^7-16510538365359116960630 x^{27} y^{58} v_1^7-7530851992287006452250 x^{26} y^{59} v_1^7-3249883100605928464776 x^{25} y^{60} v_1^7-1325020281793686766680 x^{24} y^{61} v_1^7-509592793949490565440 x^{23} y^{62} v_1^7-184543426066091279680 x^{22} y^{63} v_1^7-62802657609487339995 x^{21} y^{64} v_1^7-20039033117497187508 x^{20} y^{65} v_1^7-5979662716050990736 x^{19} y^{66} v_1^7-1663808199660842928 x^{18} y^{67} v_1^7-430226120954124180 x^{17} y^{68} v_1^7-102985550115581490 x^{16} y^{69} v_1^7-22719241748399808 x^{15} y^{70} v_1^7-4594892796281196 x^{14} y^{71} v_1^7-846727723327823 x^{13} y^{72} v_1^7-141128934022440 x^{12} y^{73} v_1^7-21089394023880 x^{11} y^{74} v_1^7-2795251491032 x^{10} y^{75} v_1^7-324267783840 x^9 y^{76} v_1^7-32374735965 x^8 y^{77} v_1^7-2721890028 x^7 y^{78} v_1^7-187187616 x^6 y^{79} v_1^7-10113642 x^5 y^{80} v_1^7-404670 x^4 y^{81} v_1^7-10920 x^3 y^{82} v_1^7-168 x^2 y^{83} v_1^7-x y^{84} v_1^7\right) d^{85}+O[d]^{86}$
$(x+y) d+\left(-x^{16} y v_1-8 x^{15} y^2 v_1-40 x^{14} y^3 v_1-140 x^{13} y^4 v_1-364 x^{12} y^5 v_1-728 x^{11} y^6 v_1-1144 x^{10} y^7 v_1-1430 x^9 y^8 v_1-1430 x^8 y^9 v_1-1144 x^7 y^{10} v_1-728 x^6 y^{11} v_1-364 x^5 y^{12} v_1-140 x^4 y^{13} v_1-40 x^3 y^{14} v_1-8 x^2 y^{15} v_1-x y^{16} v_1\right) d^{17}+\left(x^{32} y v_1^2+24 x^{31} y^2 v_1^2+288 x^{30} y^3 v_1^2+2300 x^{29} y^4 v_1^2+13704 x^{28} y^5 v_1^2+64680 x^{27} y^6 v_1^2+250624 x^{26} y^7 v_1^2+815958 x^{25} y^8 v_1^2+2267980 x^{24} y^9 v_1^2+5444296 x^{23} y^{10} v_1^2+11384256 x^{22} y^{11} v_1^2+20871500 x^{21} y^{12} v_1^2+33715640 x^{20} y^{13} v_1^2+48165240 x^{19} y^{14} v_1^2+61009312 x^{18} y^{15} v_1^2+68635477 x^{17} y^{16} v_1^2+68635477 x^{16} y^{17} v_1^2+61009312 x^{15} y^{18} v_1^2+48165240 x^{14} y^{19} v_1^2+33715640 x^{13} y^{20} v_1^2+20871500 x^{12} y^{21} v_1^2+11384256 x^{11} y^{22} v_1^2+5444296 x^{10} y^{23} v_1^2+2267980 x^9 y^{24} v_1^2+815958 x^8 y^{25} v_1^2+250624 x^7 y^{26} v_1^2+64680 x^6 y^{27} v_1^2+13704 x^5 y^{28} v_1^2+2300 x^4 y^{29} v_1^2+288 x^3 y^{30} v_1^2+24 x^2 y^{31} v_1^2+x y^{32} v_1^2\right) d^{33}+\left(-x^{48} y v_1^3-48 x^{47} y^2 v_1^3-1040 x^{46} y^3 v_1^3-14260 x^{45} y^4 v_1^3-142044 x^{44} y^5 v_1^3-1106336 x^{43} y^6 v_1^3-7046688 x^{42} y^7 v_1^3-37811070 x^{41} y^8 v_1^3-174518410 x^{40} y^9 v_1^3-703517936 x^{39} y^{10} v_1^3-2505675120 x^{38} y^{11} v_1^3-7955509380 x^{37} y^{12} v_1^3-22676319260 x^{36} y^{13} v_1^3-58358700480 x^{35} y^{14} v_1^3-136231310432 x^{34} y^{15} v_1^3-289560170145 x^{33} y^{16} v_1^3-562156024582 x^{32} y^{17} v_1^3-999449497456 x^{31} y^{18} v_1^3-1630728924208 x^{30} y^{19} v_1^3-2446127101520 x^{29} y^{20} v_1^3-3378005913280 x^{28} y^{21} v_1^3-4299291620080 x^{27} y^{22} v_1^3-5046999878896 x^{26} y^{23} v_1^3-5467585198608 x^{25} y^{24} v_1^3-5467585198608 x^{24} y^{25} v_1^3-5046999878896 x^{23} y^{26} v_1^3-4299291620080 x^{22} y^{27} v_1^3-3378005913280 x^{21} y^{28} v_1^3-2446127101520 x^{20} y^{29} v_1^3-1630728924208 x^{19} y^{30} v_1^3-999449497456 x^{18} y^{31} v_1^3-562156024582 x^{17} y^{32} v_1^3-289560170145 x^{16} y^{33} v_1^3-136231310432 x^{15} y^{34} v_1^3-58358700480 x^{14} y^{35} v_1^3-22676319260 x^{13} y^{36} v_1^3-7955509380 x^{12} y^{37} v_1^3-2505675120 x^{11} y^{38} v_1^3-703517936 x^{10} y^{39} v_1^3-174518410 x^9 y^{40} v_1^3-37811070 x^8 y^{41} v_1^3-7046688 x^7 y^{42} v_1^3-1106336 x^6 y^{43} v_1^3-142044 x^5 y^{44} v_1^3-14260 x^4 y^{45} v_1^3-1040 x^3 y^{46} v_1^3-48 x^2 y^{47} v_1^3-x y^{48} v_1^3\right) d^{49}+\left(x^{64} y v_1^4+80 x^{63} y^2 v_1^4+2720 x^{62} y^3 v_1^4+56420 x^{61} y^4 v_1^4+830368 x^{60} y^5 v_1^4+9410016 x^{59} y^6 v_1^4+86359680 x^{58} y^7 v_1^4+663918750 x^{57} y^8 v_1^4+4379337160 x^{56} y^9 v_1^4+25227806032 x^{55} y^{10} v_1^4+128644705280 x^{54} y^{11} v_1^4+586856683140 x^{53} y^{12} v_1^4+2415245873600 x^{52} y^{13} v_1^4+9029271945280 x^{51} y^{14} v_1^4+30835755924384 x^{50} y^{15} v_1^4+96651297433845 x^{49} y^{16} v_1^4+279145307451547 x^{48} y^{17} v_1^4+745386936034912 x^{47} y^{18} v_1^4+1845482623326240 x^{46} y^{19} v_1^4+4247056160749480 x^{45} y^{20} v_1^4+9104212636060180 x^{44} y^{21} v_1^4+18212724563456352 x^{43} y^{22} v_1^4+34054923355576752 x^{42} y^{23} v_1^4+59601583445126220 x^{41} y^{24} v_1^4+97752064373195454 x^{40} y^{25} v_1^4+150392838074766656 x^{39} y^{26} v_1^4+217238397716757120 x^{38} y^{27} v_1^4+294826914363810280 x^{37} y^{28} v_1^4+376160912923899220 x^{36} y^{29} v_1^4+451394699026020160 x^{35} y^{30} v_1^4+509640109880342256 x^{34} y^{31} v_1^4+541493034158120895 x^{33} y^{32} v_1^4+541493034158120895 x^{32} y^{33} v_1^4+509640109880342256 x^{31} y^{34} v_1^4+451394699026020160 x^{30} y^{35} v_1^4+376160912923899220 x^{29} y^{36} v_1^4+294826914363810280 x^{28} y^{37} v_1^4+217238397716757120 x^{27} y^{38} v_1^4+150392838074766656 x^{26} y^{39} v_1^4+97752064373195454 x^{25} y^{40} v_1^4+59601583445126220 x^{24} y^{41} v_1^4+34054923355576752 x^{23} y^{42} v_1^4+18212724563456352 x^{22} y^{43} v_1^4+9104212636060180 x^{21} y^{44} v_1^4+4247056160749480 x^{20} y^{45} v_1^4+1845482623326240 x^{19} y^{46} v_1^4+745386936034912 x^{18} y^{47} v_1^4+279145307451547 x^{17} y^{48} v_1^4+96651297433845 x^{16} y^{49} v_1^4+30835755924384 x^{15} y^{50} v_1^4+9029271945280 x^{14} y^{51} v_1^4+2415245873600 x^{13} y^{52} v_1^4+586856683140 x^{12} y^{53} v_1^4+128644705280 x^{11} y^{54} v_1^4+25227806032 x^{10} y^{55} v_1^4+4379337160 x^9 y^{56} v_1^4+663918750 x^8 y^{57} v_1^4+86359680 x^7 y^{58} v_1^4+9410016 x^6 y^{59} v_1^4+830368 x^5 y^{60} v_1^4+56420 x^4 y^{61} v_1^4+2720 x^3 y^{62} v_1^4+80 x^2 y^{63} v_1^4+x y^{64} v_1^4\right) d^{65}+\left(-x^{80} y v_1^5-120 x^{79} y^2 v_1^5-5880 x^{78} y^3 v_1^5-171080 x^{77} y^4 v_1^5-3465000 x^{76} y^5 v_1^5-53300016 x^{75} y^6 v_1^5-657431280 x^{74} y^7 v_1^5-6745158090 x^{73} y^8 v_1^5-59090063890 x^{72} y^9 v_1^5-450676266040 x^{71} y^{10} v_1^5-3037555149720 x^{70} y^{11} v_1^5-18305928389840 x^{69} y^{12} v_1^5-99577481173520 x^{68} y^{13} v_1^5-492691323359520 x^{67} y^{14} v_1^5-2231523666930240 x^{66} y^{15} v_1^5-9301686423521085 x^{65} y^{16} v_1^5-35844416926796872 x^{64} y^{17} v_1^5-128192202675757120 x^{63} y^{18} v_1^5-426903838863994320 x^{62} y^{19} v_1^5-1327648956639123440 x^{61} y^{20} v_1^5-3865608324778112000 x^{60} y^{21} v_1^5-10560780883046951712 x^{59} y^{22} v_1^5-27124753710277966320 x^{58} y^{23} v_1^5-65611089716408175600 x^{57} y^{24} v_1^5-149691036616270101072 x^{56} y^{25} v_1^5-322561856309839488928 x^{55} y^{26} v_1^5-657287686384888703760 x^{54} y^{27} v_1^5-1267921078979977235920 x^{53} y^{28} v_1^5-2317611235217592495680 x^{52} y^{29} v_1^5-4017644198223093498432 x^{51} y^{30} v_1^5-6610182338461006123248 x^{50} y^{31} v_1^5-10328951348698611556620 x^{49} y^{32} v_1^5-15337469109710473030470 x^{48} y^{33} v_1^5-21653406812672467041912 x^{47} y^{34} v_1^5-29077882399339596229560 x^{46} y^{35} v_1^5-37155445757507944831240 x^{45} y^{36} v_1^5-45189345312598966071880 x^{44} y^{37} v_1^5-52324711795476625191720 x^{43} y^{38} v_1^5-57691479214846758218472 x^{42} y^{39} v_1^5-60576115169983945969260 x^{41} y^{40} v_1^5-60576115169983945969260 x^{40} y^{41} v_1^5-57691479214846758218472 x^{39} y^{42} v_1^5-52324711795476625191720 x^{38} y^{43} v_1^5-45189345312598966071880 x^{37} y^{44} v_1^5-37155445757507944831240 x^{36} y^{45} v_1^5-29077882399339596229560 x^{35} y^{46} v_1^5-21653406812672467041912 x^{34} y^{47} v_1^5-15337469109710473030470 x^{33} y^{48} v_1^5-10328951348698611556620 x^{32} y^{49} v_1^5-6610182338461006123248 x^{31} y^{50} v_1^5-4017644198223093498432 x^{30} y^{51} v_1^5-2317611235217592495680 x^{29} y^{52} v_1^5-1267921078979977235920 x^{28} y^{53} v_1^5-657287686384888703760 x^{27} y^{54} v_1^5-322561856309839488928 x^{26} y^{55} v_1^5-149691036616270101072 x^{25} y^{56} v_1^5-65611089716408175600 x^{24} y^{57} v_1^5-27124753710277966320 x^{23} y^{58} v_1^5-10560780883046951712 x^{22} y^{59} v_1^5-3865608324778112000 x^{21} y^{60} v_1^5-1327648956639123440 x^{20} y^{61} v_1^5-426903838863994320 x^{19} y^{62} v_1^5-128192202675757120 x^{18} y^{63} v_1^5-35844416926796872 x^{17} y^{64} v_1^5-9301686423521085 x^{16} y^{65} v_1^5-2231523666930240 x^{15} y^{66} v_1^5-492691323359520 x^{14} y^{67} v_1^5-99577481173520 x^{13} y^{68} v_1^5-18305928389840 x^{12} y^{69} v_1^5-3037555149720 x^{11} y^{70} v_1^5-450676266040 x^{10} y^{71} v_1^5-59090063890 x^9 y^{72} v_1^5-6745158090 x^8 y^{73} v_1^5-657431280 x^7 y^{74} v_1^5-53300016 x^6 y^{75} v_1^5-3465000 x^5 y^{76} v_1^5-171080 x^4 y^{77} v_1^5-5880 x^3 y^{78} v_1^5-120 x^2 y^{79} v_1^5-x y^{80} v_1^5\right) d^{81}+\left(x^{96} y v_1^6+168 x^{95} y^2 v_1^6+11200 x^{94} y^3 v_1^6+434280 x^{93} y^4 v_1^6+11542608 x^{92} y^5 v_1^6+230286672 x^{91} y^6 v_1^6+3651158016 x^{90} y^7 v_1^6+47820685770 x^{89} y^8 v_1^6+531983512060 x^{88} y^9 v_1^6+5132131172168 x^{87} y^{10} v_1^6+43628047147776 x^{86} y^{11} v_1^6+330973599615568 x^{85} y^{12} v_1^6+2263635632506080 x^{84} y^{13} v_1^6+14074505118396000 x^{83} y^{14} v_1^6+80110451988721440 x^{82} y^{15} v_1^6+419867752865718465 x^{81} y^{16} v_1^6+2036390768816396617 x^{80} y^{17} v_1^6+9178817841859742080 x^{79} y^{18} v_1^6+38591462233965026680 x^{78} y^{19} v_1^6+151834351669102704560 x^{77} y^{20} v_1^6+560591564444820734760 x^{76} y^{21} v_1^6+1947149821692415701792 x^{75} y^{22} v_1^6+6376526346185372754720 x^{74} y^{23} v_1^6+19726567323785947089540 x^{73} y^{24} v_1^6+57751267622051539740906 x^{72} y^{25} v_1^6+160249149117212007825280 x^{71} y^{26} v_1^6+422053198326757279100280 x^{70} y^{27} v_1^6+1056400916888279287112320 x^{69} y^{28} v_1^6+2515823241029292135662960 x^{68} y^{29} v_1^6+5706550323638909643674496 x^{67} y^{30} v_1^6+12340122171073836085513680 x^{66} y^{31} v_1^6+25461830924585967974884965 x^{65} y^{32} v_1^6+50167428665881780195413870 x^{64} y^{33} v_1^6+94454460239824179796203320 x^{63} y^{34} v_1^6+170047106083336898413032720 x^{62} y^{35} v_1^6+292896059631836666057064040 x^{61} y^{36} v_1^6+482927880063027471249703680 x^{60} y^{37} v_1^6+762570023970878339475073320 x^{59} y^{38} v_1^6+1153689250227823819303325520 x^{58} y^{39} v_1^6+1672909949614621642032740100 x^{57} y^{40} v_1^6+2325813341681055831765485760 x^{56} y^{41} v_1^6+3101141947465907134085398056 x^{55} y^{42} v_1^6+3966628821682093042521122000 x^{54} y^{43} v_1^6+4868179754738448042424039720 x^{53} y^{44} v_1^6+5733669595475103292604400160 x^{52} y^{45} v_1^6+6481566000685032815696643320 x^{51} y^{46} v_1^6+7033205932876654464440667216 x^{50} y^{47} v_1^6+7326264631942355547384013770 x^{49} y^{48} v_1^6+7326264631942355547384013770 x^{48} y^{49} v_1^6+7033205932876654464440667216 x^{47} y^{50} v_1^6+6481566000685032815696643320 x^{46} y^{51} v_1^6+5733669595475103292604400160 x^{45} y^{52} v_1^6+4868179754738448042424039720 x^{44} y^{53} v_1^6+3966628821682093042521122000 x^{43} y^{54} v_1^6+3101141947465907134085398056 x^{42} y^{55} v_1^6+2325813341681055831765485760 x^{41} y^{56} v_1^6+1672909949614621642032740100 x^{40} y^{57} v_1^6+1153689250227823819303325520 x^{39} y^{58} v_1^6+762570023970878339475073320 x^{38} y^{59} v_1^6+482927880063027471249703680 x^{37} y^{60} v_1^6+292896059631836666057064040 x^{36} y^{61} v_1^6+170047106083336898413032720 x^{35} y^{62} v_1^6+94454460239824179796203320 x^{34} y^{63} v_1^6+50167428665881780195413870 x^{33} y^{64} v_1^6+25461830924585967974884965 x^{32} y^{65} v_1^6+12340122171073836085513680 x^{31} y^{66} v_1^6+5706550323638909643674496 x^{30} y^{67} v_1^6+2515823241029292135662960 x^{29} y^{68} v_1^6+1056400916888279287112320 x^{28} y^{69} v_1^6+422053198326757279100280 x^{27} y^{70} v_1^6+160249149117212007825280 x^{26} y^{71} v_1^6+57751267622051539740906 x^{25} y^{72} v_1^6+19726567323785947089540 x^{24} y^{73} v_1^6+6376526346185372754720 x^{23} y^{74} v_1^6+1947149821692415701792 x^{22} y^{75} v_1^6+560591564444820734760 x^{21} y^{76} v_1^6+151834351669102704560 x^{20} y^{77} v_1^6+38591462233965026680 x^{19} y^{78} v_1^6+9178817841859742080 x^{18} y^{79} v_1^6+2036390768816396617 x^{17} y^{80} v_1^6+419867752865718465 x^{16} y^{81} v_1^6+80110451988721440 x^{15} y^{82} v_1^6+14074505118396000 x^{14} y^{83} v_1^6+2263635632506080 x^{13} y^{84} v_1^6+330973599615568 x^{12} y^{85} v_1^6+43628047147776 x^{11} y^{86} v_1^6+5132131172168 x^{10} y^{87} v_1^6+531983512060 x^9 y^{88} v_1^6+47820685770 x^8 y^{89} v_1^6+3651158016 x^7 y^{90} v_1^6+230286672 x^6 y^{91} v_1^6+11542608 x^5 y^{92} v_1^6+434280 x^4 y^{93} v_1^6+11200 x^3 y^{94} v_1^6+168 x^2 y^{95} v_1^6+x y^{96} v_1^6\right) d^{97}+O[d]^{98}$
$(x+y) d+\left(-x^{18} y v_1-9 x^{17} y^2 v_1-51 x^{16} y^3 v_1-204 x^{15} y^4 v_1-612 x^{14} y^5 v_1-1428 x^{13} y^6 v_1-2652 x^{12} y^7 v_1-3978 x^{11} y^8 v_1-4862 x^{10} y^9 v_1-4862 x^9 y^{10} v_1-3978 x^8 y^{11} v_1-2652 x^7 y^{12} v_1-1428 x^6 y^{13} v_1-612 x^5 y^{14} v_1-204 x^4 y^{15} v_1-51 x^3 y^{16} v_1-9 x^2 y^{17} v_1-x y^{18} v_1\right) d^{19}+\left(x^{36} y v_1^2+27 x^{35} y^2 v_1^2+366 x^{34} y^3 v_1^2+3315 x^{33} y^4 v_1^2+22491 x^{32} y^5 v_1^2+121380 x^{31} y^6 v_1^2+540192 x^{30} y^7 v_1^2+2029698 x^{29} y^8 v_1^2+6545000 x^{28} y^9 v_1^2+18330862 x^{27} y^{10} v_1^2+44997912 x^{26} y^{11} v_1^2+97498128 x^{25} y^{12} v_1^2+187497828 x^{24} y^{13} v_1^2+321425460 x^{23} y^{14} v_1^2+492852576 x^{22} y^{15} v_1^2+677672343 x^{21} y^{16} v_1^2+837124668 x^{20} y^{17} v_1^2+930138521 x^{19} y^{18} v_1^2+930138521 x^{18} y^{19} v_1^2+837124668 x^{17} y^{20} v_1^2+677672343 x^{16} y^{21} v_1^2+492852576 x^{15} y^{22} v_1^2+321425460 x^{14} y^{23} v_1^2+187497828 x^{13} y^{24} v_1^2+97498128 x^{12} y^{25} v_1^2+44997912 x^{11} y^{26} v_1^2+18330862 x^{10} y^{27} v_1^2+6545000 x^9 y^{28} v_1^2+2029698 x^8 y^{29} v_1^2+540192 x^7 y^{30} v_1^2+121380 x^6 y^{31} v_1^2+22491 x^5 y^{32} v_1^2+3315 x^4 y^{33} v_1^2+366 x^3 y^{34} v_1^2+27 x^2 y^{35} v_1^2+x y^{36} v_1^2\right) d^{37}+\left(-x^{54} y v_1^3-54 x^{53} y^2 v_1^3-1320 x^{52} y^3 v_1^3-20475 x^{51} y^4 v_1^3-231336 x^{50} y^5 v_1^3-2049180 x^{49} y^6 v_1^3-14884452 x^{48} y^7 v_1^3-91336410 x^{47} y^8 v_1^3-483524030 x^{46} y^9 v_1^3-2242541400 x^{45} y^{10} v_1^3-9219030912 x^{44} y^{11} v_1^3-33900611472 x^{43} y^{12} v_1^3-112320289620 x^{42} y^{13} v_1^3-337282294320 x^{41} y^{14} v_1^3-922397790384 x^{40} y^{15} v_1^3-2306672148303 x^{39} y^{16} v_1^3-5292614406069 x^{38} y^{17} v_1^3-11174227218000 x^{37} y^{18} v_1^3-21761267352521 x^{36} y^{19} v_1^3-39171118359204 x^{35} y^{20} v_1^3-65285874937638 x^{34} y^{21} v_1^3-100896845028360 x^{33} y^{22} v_1^3-144765359939655 x^{32} y^{23} v_1^3-193020667387380 x^{31} y^{24} v_1^3-239345724907968 x^{30} y^{25} v_1^3-276168188499192 x^{29} y^{26} v_1^3-296625107501800 x^{28} y^{27} v_1^3-296625107501800 x^{27} y^{28} v_1^3-276168188499192 x^{26} y^{29} v_1^3-239345724907968 x^{25} y^{30} v_1^3-193020667387380 x^{24} y^{31} v_1^3-144765359939655 x^{23} y^{32} v_1^3-100896845028360 x^{22} y^{33} v_1^3-65285874937638 x^{21} y^{34} v_1^3-39171118359204 x^{20} y^{35} v_1^3-21761267352521 x^{19} y^{36} v_1^3-11174227218000 x^{18} y^{37} v_1^3-5292614406069 x^{17} y^{38} v_1^3-2306672148303 x^{16} y^{39} v_1^3-922397790384 x^{15} y^{40} v_1^3-337282294320 x^{14} y^{41} v_1^3-112320289620 x^{13} y^{42} v_1^3-33900611472 x^{12} y^{43} v_1^3-9219030912 x^{11} y^{44} v_1^3-2242541400 x^{10} y^{45} v_1^3-483524030 x^9 y^{46} v_1^3-91336410 x^8 y^{47} v_1^3-14884452 x^7 y^{48} v_1^3-2049180 x^6 y^{49} v_1^3-231336 x^5 y^{50} v_1^3-20475 x^4 y^{51} v_1^3-1320 x^3 y^{52} v_1^3-54 x^2 y^{53} v_1^3-x y^{54} v_1^3\right) d^{55}+\left(x^{72} y v_1^4+90 x^{71} y^2 v_1^4+3450 x^{70} y^3 v_1^4+80850 x^{69} y^4 v_1^4+1347066 x^{68} y^5 v_1^4+17315928 x^{67} y^6 v_1^4+180622620 x^{66} y^7 v_1^4+1581473025 x^{65} y^8 v_1^4+11905273655 x^{64} y^9 v_1^4+78436292792 x^{63} y^{10} v_1^4+458445071448 x^{62} y^{11} v_1^4+2402533480620 x^{61} y^{12} v_1^4+11385746621760 x^{60} y^{13} v_1^4+49133339244720 x^{59} y^{14} v_1^4+194180198819616 x^{58} y^{15} v_1^4+706209892869411 x^{57} y^{16} v_1^4+2373172843438800 x^{56} y^{17} v_1^4+7394378629027600 x^{55} y^{18} v_1^4+21426541509274521 x^{54} y^{19} v_1^4+57890833193400408 x^{53} y^{20} v_1^4+146170722029709960 x^{52} y^{21} v_1^4+345595330733430600 x^{51} y^{22} v_1^4+766464846551414280 x^{50} y^{23} v_1^4+1596994784315685180 x^{49} y^{24} v_1^4+3130349122979634528 x^{48} y^{25} v_1^4+5779382241354191640 x^{47} y^{26} v_1^4+10060702748787286830 x^{46} y^{27} v_1^4+16528593997320826400 x^{45} y^{28} v_1^4+25648094436413424192 x^{44} y^{29} v_1^4+37617444505610018112 x^{43} y^{30} v_1^4+52179228900780435300 x^{42} y^{31} v_1^4+68485382550213880860 x^{41} y^{32} v_1^4+85088000009881241940 x^{40} y^{33} v_1^4+100103593624090762998 x^{39} y^{34} v_1^4+111544040924813387007 x^{38} y^{35} v_1^4+117740948261921832400 x^{37} y^{36} v_1^4+117740948261921832400 x^{36} y^{37} v_1^4+111544040924813387007 x^{35} y^{38} v_1^4+100103593624090762998 x^{34} y^{39} v_1^4+85088000009881241940 x^{33} y^{40} v_1^4+68485382550213880860 x^{32} y^{41} v_1^4+52179228900780435300 x^{31} y^{42} v_1^4+37617444505610018112 x^{30} y^{43} v_1^4+25648094436413424192 x^{29} y^{44} v_1^4+16528593997320826400 x^{28} y^{45} v_1^4+10060702748787286830 x^{27} y^{46} v_1^4+5779382241354191640 x^{26} y^{47} v_1^4+3130349122979634528 x^{25} y^{48} v_1^4+1596994784315685180 x^{24} y^{49} v_1^4+766464846551414280 x^{23} y^{50} v_1^4+345595330733430600 x^{22} y^{51} v_1^4+146170722029709960 x^{21} y^{52} v_1^4+57890833193400408 x^{20} y^{53} v_1^4+21426541509274521 x^{19} y^{54} v_1^4+7394378629027600 x^{18} y^{55} v_1^4+2373172843438800 x^{17} y^{56} v_1^4+706209892869411 x^{16} y^{57} v_1^4+194180198819616 x^{15} y^{58} v_1^4+49133339244720 x^{14} y^{59} v_1^4+11385746621760 x^{13} y^{60} v_1^4+2402533480620 x^{12} y^{61} v_1^4+458445071448 x^{11} y^{62} v_1^4+78436292792 x^{10} y^{63} v_1^4+11905273655 x^9 y^{64} v_1^4+1581473025 x^8 y^{65} v_1^4+180622620 x^7 y^{66} v_1^4+17315928 x^6 y^{67} v_1^4+1347066 x^5 y^{68} v_1^4+80850 x^4 y^{69} v_1^4+3450 x^3 y^{70} v_1^4+90 x^2 y^{71} v_1^4+x y^{72} v_1^4\right) d^{73}+\left(-x^{90} y v_1^5-135 x^{89} y^2 v_1^5-7455 x^{88} y^3 v_1^5-244860 x^{87} y^4 v_1^5-5607630 x^{86} y^5 v_1^5-97691958 x^{85} y^6 v_1^5-1366882110 x^{84} y^7 v_1^5-15933735180 x^{83} y^8 v_1^5-158849720315 x^{82} y^9 v_1^5-1381003999375 x^{81} y^{10} v_1^5-10627656339573 x^{80} y^{11} v_1^5-73253575744440 x^{79} y^{12} v_1^5-456542091530280 x^{78} y^{13} v_1^5-2592724992056280 x^{77} y^{14} v_1^5-13503501824708520 x^{76} y^{15} v_1^5-64847843560234881 x^{75} y^{16} v_1^5-288466600315063275 x^{74} y^{17} v_1^5-1193312624368732175 x^{73} y^{18} v_1^5-4606259256189140246 x^{72} y^{19} v_1^5-16640424155474305290 x^{71} y^{20} v_1^5-56406652390538075160 x^{70} y^{21} v_1^5-179821307482445476950 x^{69} y^{22} v_1^5-540230387293887602580 x^{68} y^{23} v_1^5-1532249758783660075803 x^{67} y^{24} v_1^5-4109559702663142230993 x^{66} y^{25} v_1^5-10437738627462718274433 x^{65} y^{26} v_1^5-25137949991081153660590 x^{64} y^{27} v_1^5-57474700002155602853680 x^{63} y^{28} v_1^5-124884479133431637332328 x^{62} y^{29} v_1^5-258132207652650207357264 x^{61} y^{30} v_1^5-507989749121323622579664 x^{60} y^{31} v_1^5-952549264963683731301930 x^{59} y^{32} v_1^5-1703127713150266642798770 x^{58} y^{33} v_1^5-2905435613930477436374496 x^{57} y^{34} v_1^5-4731820971291874812809574 x^{56} y^{35} v_1^5-7360728137044020540853604 x^{55} y^{36} v_1^5-10941740636373405304276460 x^{54} y^{37} v_1^5-15548900838965415469442394 x^{53} y^{38} v_1^5-21130657575310364106922620 x^{52} y^{39} v_1^5-27469939745881544581618398 x^{51} y^{40} v_1^5-34169993105250767439675438 x^{50} y^{41} v_1^5-40678614487307378028365250 x^{49} y^{42} v_1^5-46354736027335600770979122 x^{48} y^{43} v_1^5-50568825172079684913072840 x^{47} y^{44} v_1^5-52816339005411327260102320 x^{46} y^{45} v_1^5-52816339005411327260102320 x^{45} y^{46} v_1^5-50568825172079684913072840 x^{44} y^{47} v_1^5-46354736027335600770979122 x^{43} y^{48} v_1^5-40678614487307378028365250 x^{42} y^{49} v_1^5-34169993105250767439675438 x^{41} y^{50} v_1^5-27469939745881544581618398 x^{40} y^{51} v_1^5-21130657575310364106922620 x^{39} y^{52} v_1^5-15548900838965415469442394 x^{38} y^{53} v_1^5-10941740636373405304276460 x^{37} y^{54} v_1^5-7360728137044020540853604 x^{36} y^{55} v_1^5-4731820971291874812809574 x^{35} y^{56} v_1^5-2905435613930477436374496 x^{34} y^{57} v_1^5-1703127713150266642798770 x^{33} y^{58} v_1^5-952549264963683731301930 x^{32} y^{59} v_1^5-507989749121323622579664 x^{31} y^{60} v_1^5-258132207652650207357264 x^{30} y^{61} v_1^5-124884479133431637332328 x^{29} y^{62} v_1^5-57474700002155602853680 x^{28} y^{63} v_1^5-25137949991081153660590 x^{27} y^{64} v_1^5-10437738627462718274433 x^{26} y^{65} v_1^5-4109559702663142230993 x^{25} y^{66} v_1^5-1532249758783660075803 x^{24} y^{67} v_1^5-540230387293887602580 x^{23} y^{68} v_1^5-179821307482445476950 x^{22} y^{69} v_1^5-56406652390538075160 x^{21} y^{70} v_1^5-16640424155474305290 x^{20} y^{71} v_1^5-4606259256189140246 x^{19} y^{72} v_1^5-1193312624368732175 x^{18} y^{73} v_1^5-288466600315063275 x^{17} y^{74} v_1^5-64847843560234881 x^{16} y^{75} v_1^5-13503501824708520 x^{15} y^{76} v_1^5-2592724992056280 x^{14} y^{77} v_1^5-456542091530280 x^{13} y^{78} v_1^5-73253575744440 x^{12} y^{79} v_1^5-10627656339573 x^{11} y^{80} v_1^5-1381003999375 x^{10} y^{81} v_1^5-158849720315 x^9 y^{82} v_1^5-15933735180 x^8 y^{83} v_1^5-1366882110 x^7 y^{84} v_1^5-97691958 x^6 y^{85} v_1^5-5607630 x^5 y^{86} v_1^5-244860 x^4 y^{87} v_1^5-7455 x^3 y^{88} v_1^5-135 x^2 y^{89} v_1^5-x y^{90} v_1^5\right) d^{91}+O[d]^{92}$
$(x+y) d+\left(-x^{22} y v_1-11 x^{21} y^2 v_1-77 x^{20} y^3 v_1-385 x^{19} y^4 v_1-1463 x^{18} y^5 v_1-4389 x^{17} y^6 v_1-10659 x^{16} y^7 v_1-21318 x^{15} y^8 v_1-35530 x^{14} y^9 v_1-49742 x^{13} y^{10} v_1-58786 x^{12} y^{11} v_1-58786 x^{11} y^{12} v_1-49742 x^{10} y^{13} v_1-35530 x^9 y^{14} v_1-21318 x^8 y^{15} v_1-10659 x^7 y^{16} v_1-4389 x^6 y^{17} v_1-1463 x^5 y^{18} v_1-385 x^4 y^{19} v_1-77 x^3 y^{20} v_1-11 x^2 y^{21} v_1-x y^{22} v_1\right) d^{23}+\left(x^{44} y v_1^2+33 x^{43} y^2 v_1^2+550 x^{42} y^3 v_1^2+6160 x^{41} y^4 v_1^2+51975 x^{40} y^5 v_1^2+350889 x^{39} y^6 v_1^2+1965612 x^{38} y^7 v_1^2+9357975 x^{37} y^8 v_1^2+38507205 x^{36} y^9 v_1^2+138675680 x^{35} y^{10} v_1^2+441299586 x^{34} y^{11} v_1^2+1250407613 x^{33} y^{12} v_1^2+3174161375 x^{32} y^{13} v_1^2+7255261530 x^{31} y^{14} v_1^2+14994228480 x^{30} y^{15} v_1^2+28114189059 x^{29} y^{16} v_1^2+47959503372 x^{28} y^{17} v_1^2+74603673375 x^{27} y^{18} v_1^2+106015746760 x^{26} y^{19} v_1^2+137820470865 x^{25} y^{20} v_1^2+164071989136 x^{24} y^{21} v_1^2+178987624513 x^{23} y^{22} v_1^2+178987624513 x^{22} y^{23} v_1^2+164071989136 x^{21} y^{24} v_1^2+137820470865 x^{20} y^{25} v_1^2+106015746760 x^{19} y^{26} v_1^2+74603673375 x^{18} y^{27} v_1^2+47959503372 x^{17} y^{28} v_1^2+28114189059 x^{16} y^{29} v_1^2+14994228480 x^{15} y^{30} v_1^2+7255261530 x^{14} y^{31} v_1^2+3174161375 x^{13} y^{32} v_1^2+1250407613 x^{12} y^{33} v_1^2+441299586 x^{11} y^{34} v_1^2+138675680 x^{10} y^{35} v_1^2+38507205 x^9 y^{36} v_1^2+9357975 x^8 y^{37} v_1^2+1965612 x^7 y^{38} v_1^2+350889 x^6 y^{39} v_1^2+51975 x^5 y^{40} v_1^2+6160 x^4 y^{41} v_1^2+550 x^3 y^{42} v_1^2+33 x^2 y^{43} v_1^2+x y^{44} v_1^2\right) d^{45}+\left(-x^{66} y v_1^3-66 x^{65} y^2 v_1^3-1980 x^{64} y^3 v_1^3-37840 x^{63} y^4 v_1^3-528759 x^{62} y^5 v_1^3-5814732 x^{61} y^6 v_1^3-52636848 x^{60} y^7 v_1^3-404134335 x^{59} y^8 v_1^3-2687832290 x^{58} y^9 v_1^3-15728102962 x^{57} y^{10} v_1^3-81941469480 x^{56} y^{11} v_1^3-383643931853 x^{55} y^{12} v_1^3-1626283103830 x^{54} y^{13} v_1^3-6280061519160 x^{53} y^{14} v_1^3-22204544929512 x^{52} y^{15} v_1^3-72192885209973 x^{51} y^{16} v_1^3-216626615133291 x^{50} y^{17} v_1^3-601815201265850 x^{49} y^{18} v_1^3-1552155745327110 x^{48} y^{19} v_1^3-3725311609255929 x^{47} y^{20} v_1^3-8337766245085739 x^{46} y^{21} v_1^3-17433690227349240 x^{45} y^{22} v_1^3-34109572910699113 x^{44} y^{23} v_1^3-62534381074937508 x^{43} y^{24} v_1^3-107559273269363322 x^{42} y^{25} v_1^3-173749701297025084 x^{41} y^{26} v_1^3-263842213610257667 x^{40} y^{27} v_1^3-376917495974082932 x^{39} y^{28} v_1^3-506889074423690772 x^{38} y^{29} v_1^3-642059509261747016 x^{37} y^{30} v_1^3-766329098943596837 x^{36} y^{31} v_1^3-862120239442387211 x^{35} y^{32} v_1^3-914369952027071358 x^{34} y^{33} v_1^3-914369952027071358 x^{33} y^{34} v_1^3-862120239442387211 x^{32} y^{35} v_1^3-766329098943596837 x^{31} y^{36} v_1^3-642059509261747016 x^{30} y^{37} v_1^3-506889074423690772 x^{29} y^{38} v_1^3-376917495974082932 x^{28} y^{39} v_1^3-263842213610257667 x^{27} y^{40} v_1^3-173749701297025084 x^{26} y^{41} v_1^3-107559273269363322 x^{25} y^{42} v_1^3-62534381074937508 x^{24} y^{43} v_1^3-34109572910699113 x^{23} y^{44} v_1^3-17433690227349240 x^{22} y^{45} v_1^3-8337766245085739 x^{21} y^{46} v_1^3-3725311609255929 x^{20} y^{47} v_1^3-1552155745327110 x^{19} y^{48} v_1^3-601815201265850 x^{18} y^{49} v_1^3-216626615133291 x^{17} y^{50} v_1^3-72192885209973 x^{16} y^{51} v_1^3-22204544929512 x^{15} y^{52} v_1^3-6280061519160 x^{14} y^{53} v_1^3-1626283103830 x^{13} y^{54} v_1^3-383643931853 x^{12} y^{55} v_1^3-81941469480 x^{11} y^{56} v_1^3-15728102962 x^{10} y^{57} v_1^3-2687832290 x^9 y^{58} v_1^3-404134335 x^8 y^{59} v_1^3-52636848 x^7 y^{60} v_1^3-5814732 x^6 y^{61} v_1^3-528759 x^5 y^{62} v_1^3-37840 x^4 y^{63} v_1^3-1980 x^3 y^{64} v_1^3-66 x^2 y^{65} v_1^3-x y^{66} v_1^3\right) d^{67}+\left(x^{88} y v_1^4+110 x^{87} y^2 v_1^4+5170 x^{86} y^3 v_1^4+148995 x^{85} y^4 v_1^4+3061674 x^{84} y^5 v_1^4+48678168 x^{83} y^6 v_1^4+629820840 x^{82} y^7 v_1^4+6859797945 x^{81} y^8 v_1^4+64426013795 x^{80} y^9 v_1^4+531136213322 x^{79} y^{10} v_1^4+3896465183338 x^{78} y^{11} v_1^4+25710667623550 x^{77} y^{12} v_1^4+153912545181780 x^{76} y^{13} v_1^4+841805306791680 x^{75} y^{14} v_1^4+4231231078887912 x^{74} y^{15} v_1^4+19641636625066566 x^{73} y^{16} v_1^4+84560125063948545 x^{72} y^{17} v_1^4+338842315457060030 x^{71} y^{18} v_1^4+1267752387190130380 x^{70} y^{19} v_1^4+4440858666774712259 x^{69} y^{20} v_1^4+14599730528504854590 x^{68} y^{21} v_1^4+45143873505605990700 x^{67} y^{22} v_1^4+131540175871849889413 x^{66} y^{23} v_1^4+361798018028662133391 x^{65} y^{24} v_1^4+940782406147790909967 x^{64} y^{25} v_1^4+2315945826372782337052 x^{63} y^{26} v_1^4+5404137437083435622495 x^{62} y^{27} v_1^4+11966681242466437504776 x^{61} y^{28} v_1^4+25171801916331400693968 x^{60} y^{29} v_1^4+50344245892171957861256 x^{59} y^{30} v_1^4+95817234317425449077106 x^{58} y^{31} v_1^4+173669599320568197143440 x^{57} y^{32} v_1^4+299975676832724473595820 x^{56} y^{33} v_1^4+494078499741363256455918 x^{55} y^{34} v_1^4+776409933141778833496456 x^{54} y^{35} v_1^4+1164615666039327769185776 x^{53} y^{36} v_1^4+1668233893404145167323520 x^{52} y^{37} v_1^4+2282846887306466854335204 x^{51} y^{38} v_1^4+2985261690993085010631311 x^{50} y^{39} v_1^4+3731577377312786604630192 x^{49} y^{40} v_1^4+4459690209331106144948322 x^{48} y^{41} v_1^4+5096788916449502297501850 x^{47} y^{42} v_1^4+5570908874116636013252817 x^{46} y^{43} v_1^4+5824132030151118577419240 x^{45} y^{44} v_1^4+5824132030151118577419240 x^{44} y^{45} v_1^4+5570908874116636013252817 x^{43} y^{46} v_1^4+5096788916449502297501850 x^{42} y^{47} v_1^4+4459690209331106144948322 x^{41} y^{48} v_1^4+3731577377312786604630192 x^{40} y^{49} v_1^4+2985261690993085010631311 x^{39} y^{50} v_1^4+2282846887306466854335204 x^{38} y^{51} v_1^4+1668233893404145167323520 x^{37} y^{52} v_1^4+1164615666039327769185776 x^{36} y^{53} v_1^4+776409933141778833496456 x^{35} y^{54} v_1^4+494078499741363256455918 x^{34} y^{55} v_1^4+299975676832724473595820 x^{33} y^{56} v_1^4+173669599320568197143440 x^{32} y^{57} v_1^4+95817234317425449077106 x^{31} y^{58} v_1^4+50344245892171957861256 x^{30} y^{59} v_1^4+25171801916331400693968 x^{29} y^{60} v_1^4+11966681242466437504776 x^{28} y^{61} v_1^4+5404137437083435622495 x^{27} y^{62} v_1^4+2315945826372782337052 x^{26} y^{63} v_1^4+940782406147790909967 x^{25} y^{64} v_1^4+361798018028662133391 x^{24} y^{65} v_1^4+131540175871849889413 x^{23} y^{66} v_1^4+45143873505605990700 x^{22} y^{67} v_1^4+14599730528504854590 x^{21} y^{68} v_1^4+4440858666774712259 x^{20} y^{69} v_1^4+1267752387190130380 x^{19} y^{70} v_1^4+338842315457060030 x^{18} y^{71} v_1^4+84560125063948545 x^{17} y^{72} v_1^4+19641636625066566 x^{16} y^{73} v_1^4+4231231078887912 x^{15} y^{74} v_1^4+841805306791680 x^{14} y^{75} v_1^4+153912545181780 x^{13} y^{76} v_1^4+25710667623550 x^{12} y^{77} v_1^4+3896465183338 x^{11} y^{78} v_1^4+531136213322 x^{10} y^{79} v_1^4+64426013795 x^9 y^{80} v_1^4+6859797945 x^8 y^{81} v_1^4+629820840 x^7 y^{82} v_1^4+48678168 x^6 y^{83} v_1^4+3061674 x^5 y^{84} v_1^4+148995 x^4 y^{85} v_1^4+5170 x^3 y^{86} v_1^4+110 x^2 y^{87} v_1^4+x y^{88} v_1^4\right) d^{89}+\left(-x^{110} y v_1^5-165 x^{109} y^2 v_1^5-11165 x^{108} y^3 v_1^5-450450 x^{107} y^4 v_1^5-12701304 x^{106} y^5 v_1^5-273067872 x^{105} y^6 v_1^5-4725838920 x^{104} y^7 v_1^5-68295703905 x^{103} y^8 v_1^5-846032402930 x^{102} y^9 v_1^5-9160666723208 x^{101} y^{10} v_1^5-88008041460066 x^{100} y^{11} v_1^5-759111013124100 x^{99} y^{12} v_1^5-5934834875896080 x^{98} y^{13} v_1^5-42385649438064240 x^{97} y^{14} v_1^5-278325097445036664 x^{96} y^{15} v_1^5-1689592221295286550 x^{95} y^{16} v_1^5-9526399008772902795 x^{94} y^{17} v_1^5-50087814916826663515 x^{93} y^{18} v_1^5-246434425401131167585 x^{92} y^{19} v_1^5-1138039215511978083150 x^{91} y^{20} v_1^5-4946102997747076548240 x^{90} y^{21} v_1^5-20279201591561828233500 x^{89} y^{22} v_1^5-78603233291045880879913 x^{88} y^{23} v_1^5-288573653418530225359735 x^{87} y^{24} v_1^5-1005177096302632975161462 x^{86} y^{25} v_1^5-3327132495135081854007864 x^{85} y^{26} v_1^5-10479710140640118901474490 x^{84} y^{27} v_1^5-31451097103162823132743224 x^{83} y^{28} v_1^5-90040380752347859537845680 x^{82} y^{29} v_1^5-246160718302309652973129152 x^{81} y^{30} v_1^5-643289952153255533035523163 x^{80} y^{31} v_1^5-1608398549982459239720916860 x^{79} y^{32} v_1^5-3850708625634839935933946714 x^{78} y^{33} v_1^5-8834472690250247571449501310 x^{77} y^{34} v_1^5-19436616328483629872553627528 x^{76} y^{35} v_1^5-41034021309131155243342571244 x^{75} y^{36} v_1^5-83178738455103498951996606420 x^{74} y^{37} v_1^5-161981931417343775186536208088 x^{73} y^{38} v_1^5-303199933812066935370741742365 x^{72} y^{39} v_1^5-545763612438945588228824659068 x^{71} y^{40} v_1^5-945107300742430769151244936900 x^{70} y^{41} v_1^5-1575183931357521477442388846050 x^{69} y^{42} v_1^5-2527626297963817435136417327970 x^{68} y^{43} v_1^5-3906337375508093876421714878100 x^{67} y^{44} v_1^5-5816108138710200155460339540480 x^{66} y^{45} v_1^5-8344856378434777565610802752174 x^{65} y^{46} v_1^5-11540763917528143483392491974635 x^{64} y^{47} v_1^5-15387689681806690767432413034546 x^{63} y^{48} v_1^5-19784176176636906522720415526970 x^{62} y^{49} v_1^5-24532381437590324659274865712860 x^{61} y^{50} v_1^5-29342654576239872569708307682128 x^{60} y^{51} v_1^5-33856910765620128773426213848320 x^{59} y^{52} v_1^5-37689769696715233775063639058168 x^{58} y^{53} v_1^5-40481605162559636262945778290930 x^{57} y^{54} v_1^5-41953663846200901663463658736044 x^{56} y^{55} v_1^5-41953663846200901663463658736044 x^{55} y^{56} v_1^5-40481605162559636262945778290930 x^{54} y^{57} v_1^5-37689769696715233775063639058168 x^{53} y^{58} v_1^5-33856910765620128773426213848320 x^{52} y^{59} v_1^5-29342654576239872569708307682128 x^{51} y^{60} v_1^5-24532381437590324659274865712860 x^{50} y^{61} v_1^5-19784176176636906522720415526970 x^{49} y^{62} v_1^5-15387689681806690767432413034546 x^{48} y^{63} v_1^5-11540763917528143483392491974635 x^{47} y^{64} v_1^5-8344856378434777565610802752174 x^{46} y^{65} v_1^5-5816108138710200155460339540480 x^{45} y^{66} v_1^5-3906337375508093876421714878100 x^{44} y^{67} v_1^5-2527626297963817435136417327970 x^{43} y^{68} v_1^5-1575183931357521477442388846050 x^{42} y^{69} v_1^5-945107300742430769151244936900 x^{41} y^{70} v_1^5-545763612438945588228824659068 x^{40} y^{71} v_1^5-303199933812066935370741742365 x^{39} y^{72} v_1^5-161981931417343775186536208088 x^{38} y^{73} v_1^5-83178738455103498951996606420 x^{37} y^{74} v_1^5-41034021309131155243342571244 x^{36} y^{75} v_1^5-19436616328483629872553627528 x^{35} y^{76} v_1^5-8834472690250247571449501310 x^{34} y^{77} v_1^5-3850708625634839935933946714 x^{33} y^{78} v_1^5-1608398549982459239720916860 x^{32} y^{79} v_1^5-643289952153255533035523163 x^{31} y^{80} v_1^5-246160718302309652973129152 x^{30} y^{81} v_1^5-90040380752347859537845680 x^{29} y^{82} v_1^5-31451097103162823132743224 x^{28} y^{83} v_1^5-10479710140640118901474490 x^{27} y^{84} v_1^5-3327132495135081854007864 x^{26} y^{85} v_1^5-1005177096302632975161462 x^{25} y^{86} v_1^5-288573653418530225359735 x^{24} y^{87} v_1^5-78603233291045880879913 x^{23} y^{88} v_1^5-20279201591561828233500 x^{22} y^{89} v_1^5-4946102997747076548240 x^{21} y^{90} v_1^5-1138039215511978083150 x^{20} y^{91} v_1^5-246434425401131167585 x^{19} y^{92} v_1^5-50087814916826663515 x^{18} y^{93} v_1^5-9526399008772902795 x^{17} y^{94} v_1^5-1689592221295286550 x^{16} y^{95} v_1^5-278325097445036664 x^{15} y^{96} v_1^5-42385649438064240 x^{14} y^{97} v_1^5-5934834875896080 x^{13} y^{98} v_1^5-759111013124100 x^{12} y^{99} v_1^5-88008041460066 x^{11} y^{100} v_1^5-9160666723208 x^{10} y^{101} v_1^5-846032402930 x^9 y^{102} v_1^5-68295703905 x^8 y^{103} v_1^5-4725838920 x^7 y^{104} v_1^5-273067872 x^6 y^{105} v_1^5-12701304 x^5 y^{106} v_1^5-450450 x^4 y^{107} v_1^5-11165 x^3 y^{108} v_1^5-165 x^2 y^{109} v_1^5-x y^{110} v_1^5\right) d^{111}+O[d]^{112}$
$(x+y) d+\left(-x^{28} y v_1-14 x^{27} y^2 v_1-126 x^{26} y^3 v_1-819 x^{25} y^4 v_1-4095 x^{24} y^5 v_1-16380 x^{23} y^6 v_1-53820 x^{22} y^7 v_1-148005 x^{21} y^8 v_1-345345 x^{20} y^9 v_1-690690 x^{19} y^{10} v_1-1193010 x^{18} y^{11} v_1-1789515 x^{17} y^{12} v_1-2340135 x^{16} y^{13} v_1-2674440 x^{15} y^{14} v_1-2674440 x^{14} y^{15} v_1-2340135 x^{13} y^{16} v_1-1789515 x^{12} y^{17} v_1-1193010 x^{11} y^{18} v_1-690690 x^{10} y^{19} v_1-345345 x^9 y^{20} v_1-148005 x^8 y^{21} v_1-53820 x^7 y^{22} v_1-16380 x^6 y^{23} v_1-4095 x^5 y^{24} v_1-819 x^4 y^{25} v_1-126 x^3 y^{26} v_1-14 x^2 y^{27} v_1-x y^{28} v_1\right) d^{29}+\left(x^{56} y v_1^2+42 x^{55} y^2 v_1^2+896 x^{54} y^3 v_1^2+12915 x^{53} y^4 v_1^2+140994 x^{52} y^5 v_1^2+1238328 x^{51} y^6 v_1^2+9075924 x^{50} y^7 v_1^2+56872530 x^{49} y^8 v_1^2+309984675 x^{48} y^9 v_1^2+1488617130 x^{47} y^{10} v_1^2+6361648020 x^{46} y^{11} v_1^2+24388106925 x^{45} y^{12} v_1^2+84422710260 x^{44} y^{13} v_1^2+265331192400 x^{43} y^{14} v_1^2+760618759320 x^{42} y^{15} v_1^2+1996626583350 x^{41} y^{16} v_1^2+4815395314065 x^{40} y^{17} v_1^2+10700879668710 x^{39} y^{18} v_1^2+21964964221200 x^{38} y^{19} v_1^2+41733432365625 x^{37} y^{20} v_1^2+73530333363630 x^{36} y^{21} v_1^2+120322363739760 x^{35} y^{22} v_1^2+183099249185580 x^{34} y^{23} v_1^2+259390603017000 x^{33} y^{24} v_1^2+342395595983259 x^{32} y^{25} v_1^2+421409964287214 x^{31} y^{26} v_1^2+483841070107556 x^{30} y^{27} v_1^2+518401146543811 x^{29} y^{28} v_1^2+518401146543811 x^{28} y^{29} v_1^2+483841070107556 x^{27} y^{30} v_1^2+421409964287214 x^{26} y^{31} v_1^2+342395595983259 x^{25} y^{32} v_1^2+259390603017000 x^{24} y^{33} v_1^2+183099249185580 x^{23} y^{34} v_1^2+120322363739760 x^{22} y^{35} v_1^2+73530333363630 x^{21} y^{36} v_1^2+41733432365625 x^{20} y^{37} v_1^2+21964964221200 x^{19} y^{38} v_1^2+10700879668710 x^{18} y^{39} v_1^2+4815395314065 x^{17} y^{40} v_1^2+1996626583350 x^{16} y^{41} v_1^2+760618759320 x^{15} y^{42} v_1^2+265331192400 x^{14} y^{43} v_1^2+84422710260 x^{13} y^{44} v_1^2+24388106925 x^{12} y^{45} v_1^2+6361648020 x^{11} y^{46} v_1^2+1488617130 x^{10} y^{47} v_1^2+309984675 x^9 y^{48} v_1^2+56872530 x^8 y^{49} v_1^2+9075924 x^7 y^{50} v_1^2+1238328 x^6 y^{51} v_1^2+140994 x^5 y^{52} v_1^2+12915 x^4 y^{53} v_1^2+896 x^3 y^{54} v_1^2+42 x^2 y^{55} v_1^2+x y^{56} v_1^2\right) d^{57}+\left(-x^{84} y v_1^3-84 x^{83} y^2 v_1^3-3220 x^{82} y^3 v_1^3-78925 x^{81} y^4 v_1^3-1419579 x^{80} y^5 v_1^3-20166048 x^{79} y^6 v_1^3-236664180 x^{78} y^7 v_1^3-2364348285 x^{77} y^8 v_1^3-20538297780 x^{76} y^9 v_1^3-157579680258 x^{75} y^{10} v_1^3-1080768558870 x^{74} y^{11} v_1^3-6689127553290 x^{73} y^{12} v_1^3-37646446663350 x^{72} y^{13} v_1^3-193875628318200 x^{71} y^{14} v_1^3-918438592798800 x^{70} y^{15} v_1^3-4020165470078100 x^{69} y^{16} v_1^3-16321957597395765 x^{68} y^{17} v_1^3-61671429580941600 x^{67} y^{18} v_1^3-217494900854910000 x^{66} y^{19} v_1^3-717774906253568625 x^{65} y^{20} v_1^3-2221757763975361755 x^{64} y^{21} v_1^3-6463415635746610320 x^{63} y^{22} v_1^3-17704321579772509500 x^{62} y^{23} v_1^3-45736423471681999875 x^{61} y^{24} v_1^3-111597215666500062954 x^{60} y^{25} v_1^3-257532457563425970954 x^{59} y^{26} v_1^3-562756594813000932974 x^{58} y^{27} v_1^3-1165710607656648476400 x^{57} y^{28} v_1^3-2291224816209041825011 x^{56} y^{29} v_1^3-4276953474097948180908 x^{55} y^{30} v_1^3-7588143681906323962944 x^{54} y^{31} v_1^3-12804992805612517669215 x^{53} y^{32} v_1^3-20565594765374343495300 x^{52} y^{33} v_1^3-31453262765436480198048 x^{51} y^{34} v_1^3-45831897292815518795352 x^{50} y^{35} v_1^3-63655412980218541307280 x^{49} y^{36} v_1^3-84300411826347100941375 x^{48} y^{37} v_1^3-106484730749981963324400 x^{47} y^{38} v_1^3-128327752452985041495420 x^{46} y^{39} v_1^3-147576915325740877138575 x^{45} y^{40} v_1^3-161974663164368382837600 x^{44} y^{41} v_1^3-169687742363343815083200 x^{43} y^{42} v_1^3-169687742363343815083200 x^{42} y^{43} v_1^3-161974663164368382837600 x^{41} y^{44} v_1^3-147576915325740877138575 x^{40} y^{45} v_1^3-128327752452985041495420 x^{39} y^{46} v_1^3-106484730749981963324400 x^{38} y^{47} v_1^3-84300411826347100941375 x^{37} y^{48} v_1^3-63655412980218541307280 x^{36} y^{49} v_1^3-45831897292815518795352 x^{35} y^{50} v_1^3-31453262765436480198048 x^{34} y^{51} v_1^3-20565594765374343495300 x^{33} y^{52} v_1^3-12804992805612517669215 x^{32} y^{53} v_1^3-7588143681906323962944 x^{31} y^{54} v_1^3-4276953474097948180908 x^{30} y^{55} v_1^3-2291224816209041825011 x^{29} y^{56} v_1^3-1165710607656648476400 x^{28} y^{57} v_1^3-562756594813000932974 x^{27} y^{58} v_1^3-257532457563425970954 x^{26} y^{59} v_1^3-111597215666500062954 x^{25} y^{60} v_1^3-45736423471681999875 x^{24} y^{61} v_1^3-17704321579772509500 x^{23} y^{62} v_1^3-6463415635746610320 x^{22} y^{63} v_1^3-2221757763975361755 x^{21} y^{64} v_1^3-717774906253568625 x^{20} y^{65} v_1^3-217494900854910000 x^{19} y^{66} v_1^3-61671429580941600 x^{18} y^{67} v_1^3-16321957597395765 x^{17} y^{68} v_1^3-4020165470078100 x^{16} y^{69} v_1^3-918438592798800 x^{15} y^{70} v_1^3-193875628318200 x^{14} y^{71} v_1^3-37646446663350 x^{13} y^{72} v_1^3-6689127553290 x^{12} y^{73} v_1^3-1080768558870 x^{11} y^{74} v_1^3-157579680258 x^{10} y^{75} v_1^3-20538297780 x^9 y^{76} v_1^3-2364348285 x^8 y^{77} v_1^3-236664180 x^7 y^{78} v_1^3-20166048 x^6 y^{79} v_1^3-1419579 x^5 y^{80} v_1^3-78925 x^4 y^{81} v_1^3-3220 x^3 y^{82} v_1^3-84 x^2 y^{83} v_1^3-x y^{84} v_1^3\right) d^{85}+\left(x^{112} y v_1^4+140 x^{111} y^2 v_1^4+8400 x^{110} y^3 v_1^4+309925 x^{109} y^4 v_1^4+8175944 x^{108} y^5 v_1^4+167333040 x^{107} y^6 v_1^4+2794469220 x^{106} y^7 v_1^4+39391065450 x^{105} y^8 v_1^4+480100728030 x^{104} y^9 v_1^4+5150627251770 x^{103} y^{10} v_1^4+49309369189080 x^{102} y^{11} v_1^4+425818765660470 x^{101} y^{12} v_1^4+3345930702948540 x^{100} y^{13} v_1^4+24093380649379200 x^{99} y^{14} v_1^4+159934750878701520 x^{98} y^{15} v_1^4+983620514602124910 x^{97} y^{16} v_1^4+5628744893856579075 x^{96} y^{17} v_1^4+30081644196816030000 x^{95} y^{18} v_1^4+150625715884935060000 x^{94} y^{19} v_1^4+708658639565448350625 x^{93} y^{20} v_1^4+3140567161553818057380 x^{92} y^{21} v_1^4+13139744273042622123000 x^{91} y^{22} v_1^4+52005388184487538300500 x^{90} y^{23} v_1^4+195065942115299950626750 x^{89} y^{24} v_1^4+694546351146134324294184 x^{88} y^{25} v_1^4+2351029797875248831274346 x^{87} y^{26} v_1^4+7576103216415059235039200 x^{86} y^{27} v_1^4+23270625589596767156096800 x^{85} y^{28} v_1^4+68209297263289492085557011 x^{84} y^{29} v_1^4+190990309290684675787740536 x^{83} y^{30} v_1^4+511368738825192489884687380 x^{82} y^{31} v_1^4+1310395198232361367847172375 x^{81} y^{32} v_1^4+3216445143074197822695451950 x^{80} y^{33} v_1^4+7568137672260877960466156568 x^{79} y^{34} v_1^4+17082413720714703069096888240 x^{78} y^{35} v_1^4+37011960050294836867738459410 x^{77} y^{36} v_1^4+77024974134809189553420037500 x^{76} y^{37} v_1^4+154050054754349129047726803840 x^{75} y^{38} v_1^4+296250233624577700850709040770 x^{74} y^{39} v_1^4+548063079782384070315267030090 x^{73} y^{40} v_1^4+975819791831103082531997028000 x^{72} y^{41} v_1^4+1672834098541061868861901422600 x^{71} y^{42} v_1^4+2762121588209030245297893930000 x^{70} y^{43} v_1^4+4394284506852664366189998273600 x^{69} y^{44} v_1^4+6737903058084327856311820371675 x^{68} y^{45} v_1^4+9960378562017604286378240677200 x^{67} y^{46} v_1^4+14198837631062929796823669756000 x^{66} y^{47} v_1^4+19523401827011641241490971354625 x^{65} y^{48} v_1^4+25898390242343372680790105717280 x^{64} y^{49} v_1^4+33149939556028570474288965650424 x^{63} y^{50} v_1^4+40949925365362924426125750777000 x^{62} y^{51} v_1^4+48824911033092587966948394506550 x^{61} y^{52} v_1^4+56194708937632501185840432063690 x^{60} y^{53} v_1^4+62438565493722481426544026185984 x^{59} y^{54} v_1^4+66979552079084807822640773866122 x^{58} y^{55} v_1^4+69371678940760492159316378077200 x^{57} y^{56} v_1^4+69371678940760492159316378077200 x^{56} y^{57} v_1^4+66979552079084807822640773866122 x^{55} y^{58} v_1^4+62438565493722481426544026185984 x^{54} y^{59} v_1^4+56194708937632501185840432063690 x^{53} y^{60} v_1^4+48824911033092587966948394506550 x^{52} y^{61} v_1^4+40949925365362924426125750777000 x^{51} y^{62} v_1^4+33149939556028570474288965650424 x^{50} y^{63} v_1^4+25898390242343372680790105717280 x^{49} y^{64} v_1^4+19523401827011641241490971354625 x^{48} y^{65} v_1^4+14198837631062929796823669756000 x^{47} y^{66} v_1^4+9960378562017604286378240677200 x^{46} y^{67} v_1^4+6737903058084327856311820371675 x^{45} y^{68} v_1^4+4394284506852664366189998273600 x^{44} y^{69} v_1^4+2762121588209030245297893930000 x^{43} y^{70} v_1^4+1672834098541061868861901422600 x^{42} y^{71} v_1^4+975819791831103082531997028000 x^{41} y^{72} v_1^4+548063079782384070315267030090 x^{40} y^{73} v_1^4+296250233624577700850709040770 x^{39} y^{74} v_1^4+154050054754349129047726803840 x^{38} y^{75} v_1^4+77024974134809189553420037500 x^{37} y^{76} v_1^4+37011960050294836867738459410 x^{36} y^{77} v_1^4+17082413720714703069096888240 x^{35} y^{78} v_1^4+7568137672260877960466156568 x^{34} y^{79} v_1^4+3216445143074197822695451950 x^{33} y^{80} v_1^4+1310395198232361367847172375 x^{32} y^{81} v_1^4+511368738825192489884687380 x^{31} y^{82} v_1^4+190990309290684675787740536 x^{30} y^{83} v_1^4+68209297263289492085557011 x^{29} y^{84} v_1^4+23270625589596767156096800 x^{28} y^{85} v_1^4+7576103216415059235039200 x^{27} y^{86} v_1^4+2351029797875248831274346 x^{26} y^{87} v_1^4+694546351146134324294184 x^{25} y^{88} v_1^4+195065942115299950626750 x^{24} y^{89} v_1^4+52005388184487538300500 x^{23} y^{90} v_1^4+13139744273042622123000 x^{22} y^{91} v_1^4+3140567161553818057380 x^{21} y^{92} v_1^4+708658639565448350625 x^{20} y^{93} v_1^4+150625715884935060000 x^{19} y^{94} v_1^4+30081644196816030000 x^{18} y^{95} v_1^4+5628744893856579075 x^{17} y^{96} v_1^4+983620514602124910 x^{16} y^{97} v_1^4+159934750878701520 x^{15} y^{98} v_1^4+24093380649379200 x^{14} y^{99} v_1^4+3345930702948540 x^{13} y^{100} v_1^4+425818765660470 x^{12} y^{101} v_1^4+49309369189080 x^{11} y^{102} v_1^4+5150627251770 x^{10} y^{103} v_1^4+480100728030 x^9 y^{104} v_1^4+39391065450 x^8 y^{105} v_1^4+2794469220 x^7 y^{106} v_1^4+167333040 x^6 y^{107} v_1^4+8175944 x^5 y^{108} v_1^4+309925 x^4 y^{109} v_1^4+8400 x^3 y^{110} v_1^4+140 x^2 y^{111} v_1^4+x y^{112} v_1^4\right) d^{113}+O[d]^{114}$
$(x+y) d+\left(-x^{30} y v_1-15 x^{29} y^2 v_1-145 x^{28} y^3 v_1-1015 x^{27} y^4 v_1-5481 x^{26} y^5 v_1-23751 x^{25} y^6 v_1-84825 x^{24} y^7 v_1-254475 x^{23} y^8 v_1-650325 x^{22} y^9 v_1-1430715 x^{21} y^{10} v_1-2731365 x^{20} y^{11} v_1-4552275 x^{19} y^{12} v_1-6653325 x^{18} y^{13} v_1-8554275 x^{17} y^{14} v_1-9694845 x^{16} y^{15} v_1-9694845 x^{15} y^{16} v_1-8554275 x^{14} y^{17} v_1-6653325 x^{13} y^{18} v_1-4552275 x^{12} y^{19} v_1-2731365 x^{11} y^{20} v_1-1430715 x^{10} y^{21} v_1-650325 x^9 y^{22} v_1-254475 x^8 y^{23} v_1-84825 x^7 y^{24} v_1-23751 x^6 y^{25} v_1-5481 x^5 y^{26} v_1-1015 x^4 y^{27} v_1-145 x^3 y^{28} v_1-15 x^2 y^{29} v_1-x y^{30} v_1\right) d^{31}+\left(x^{60} y v_1^2+45 x^{59} y^2 v_1^2+1030 x^{58} y^3 v_1^2+15950 x^{57} y^4 v_1^2+187311 x^{56} y^5 v_1^2+1771987 x^{55} y^6 v_1^2+14007580 x^{54} y^7 v_1^2+94805640 x^{53} y^8 v_1^2+558950205 x^{52} y^9 v_1^2+2907971781 x^{51} y^{10} v_1^2+13485145986 x^{50} y^{11} v_1^2+56192660550 x^{49} y^{12} v_1^2+211809758475 x^{48} y^{13} v_1^2+726213440475 x^{47} y^{14} v_1^2+2275478475000 x^{46} y^{15} v_1^2+6542010310470 x^{45} y^{16} v_1^2+17317094670225 x^{44} y^{17} v_1^2+42330682513875 x^{43} y^{18} v_1^2+95801022873150 x^{42} y^{19} v_1^2+201182150764980 x^{41} y^{20} v_1^2+392784200543295 x^{40} y^{21} v_1^2+714153092547225 x^{39} y^{22} v_1^2+1210955244138900 x^{38} y^{23} v_1^2+1917345803304750 x^{37} y^{24} v_1^2+2837671788914781 x^{36} y^{25} v_1^2+3929084015425947 x^{35} y^{26} v_1^2+5093257057034650 x^{34} y^{27} v_1^2+6184669283542220 x^{33} y^{28} v_1^2+7037727115754955 x^{32} y^{29} v_1^2+7506908923471953 x^{31} y^{30} v_1^2+7506908923471953 x^{30} y^{31} v_1^2+7037727115754955 x^{29} y^{32} v_1^2+6184669283542220 x^{28} y^{33} v_1^2+5093257057034650 x^{27} y^{34} v_1^2+3929084015425947 x^{26} y^{35} v_1^2+2837671788914781 x^{25} y^{36} v_1^2+1917345803304750 x^{24} y^{37} v_1^2+1210955244138900 x^{23} y^{38} v_1^2+714153092547225 x^{22} y^{39} v_1^2+392784200543295 x^{21} y^{40} v_1^2+201182150764980 x^{20} y^{41} v_1^2+95801022873150 x^{19} y^{42} v_1^2+42330682513875 x^{18} y^{43} v_1^2+17317094670225 x^{17} y^{44} v_1^2+6542010310470 x^{16} y^{45} v_1^2+2275478475000 x^{15} y^{46} v_1^2+726213440475 x^{14} y^{47} v_1^2+211809758475 x^{13} y^{48} v_1^2+56192660550 x^{12} y^{49} v_1^2+13485145986 x^{11} y^{50} v_1^2+2907971781 x^{10} y^{51} v_1^2+558950205 x^9 y^{52} v_1^2+94805640 x^8 y^{53} v_1^2+14007580 x^7 y^{54} v_1^2+1771987 x^6 y^{55} v_1^2+187311 x^5 y^{56} v_1^2+15950 x^4 y^{57} v_1^2+1030 x^3 y^{58} v_1^2+45 x^2 y^{59} v_1^2+x y^{60} v_1^2\right) d^{61}+\left(-x^{90} y v_1^3-90 x^{89} y^2 v_1^3-3700 x^{88} y^3 v_1^3-97350 x^{87} y^4 v_1^3-1881201 x^{86} y^5 v_1^3-28735868 x^{85} y^6 v_1^3-362943120 x^{84} y^7 v_1^3-3905708400 x^{83} y^8 v_1^3-36578261005 x^{82} y^9 v_1^3-302849712022 x^{81} y^{10} v_1^3-2243560298148 x^{80} y^{11} v_1^3-15013261314870 x^{79} y^{12} v_1^3-91446243902685 x^{78} y^{13} v_1^3-510212429469720 x^{77} y^{14} v_1^3-2621365949752896 x^{76} y^{15} v_1^3-12458030271636726 x^{75} y^{16} v_1^3-54979215351891075 x^{74} y^{17} v_1^3-226067993795843850 x^{73} y^{18} v_1^3-868672829817431100 x^{72} y^{19} v_1^3-3127423369493516940 x^{71} y^{20} v_1^3-10574062271535767235 x^{70} y^{21} v_1^3-33645457744342715700 x^{69} y^{22} v_1^3-100937584188272286000 x^{68} y^{23} v_1^3-285991739212574781750 x^{67} y^{24} v_1^3-766460698761489329871 x^{66} y^{25} v_1^3-1945634933632411417158 x^{65} y^{26} v_1^3-4683941044594343779660 x^{64} y^{27} v_1^3-10706157143742069324300 x^{63} y^{28} v_1^3-23258210487925404287055 x^{62} y^{29} v_1^3-48066975848621425665200 x^{61} y^{30} v_1^3-94583411596131728813153 x^{60} y^{31} v_1^3-177343903780474107279615 x^{59} y^{32} v_1^3-317069409913395717769330 x^{58} y^{33} v_1^3-540883116121990928521750 x^{57} y^{34} v_1^3-880866793042040670421107 x^{56} y^{35} v_1^3-1370237236458623942611797 x^{55} y^{36} v_1^3-2036839137193678688336900 x^{54} y^{37} v_1^3-2894455617223024939238460 x^{53} y^{38} v_1^3-3933490967709545829853365 x^{52} y^{39} v_1^3-5113538258415193052717403 x^{51} y^{40} v_1^3-6360742711888369891790022 x^{50} y^{41} v_1^3-7572312752343844364116050 x^{49} y^{42} v_1^3-8628914531782926505800375 x^{48} y^{43} v_1^3-9413361307416642217625025 x^{47} y^{44} v_1^3-9831732921085387614681000 x^{46} y^{45} v_1^3-9831732921085387614681000 x^{45} y^{46} v_1^3-9413361307416642217625025 x^{44} y^{47} v_1^3-8628914531782926505800375 x^{43} y^{48} v_1^3-7572312752343844364116050 x^{42} y^{49} v_1^3-6360742711888369891790022 x^{41} y^{50} v_1^3-5113538258415193052717403 x^{40} y^{51} v_1^3-3933490967709545829853365 x^{39} y^{52} v_1^3-2894455617223024939238460 x^{38} y^{53} v_1^3-2036839137193678688336900 x^{37} y^{54} v_1^3-1370237236458623942611797 x^{36} y^{55} v_1^3-880866793042040670421107 x^{35} y^{56} v_1^3-540883116121990928521750 x^{34} y^{57} v_1^3-317069409913395717769330 x^{33} y^{58} v_1^3-177343903780474107279615 x^{32} y^{59} v_1^3-94583411596131728813153 x^{31} y^{60} v_1^3-48066975848621425665200 x^{30} y^{61} v_1^3-23258210487925404287055 x^{29} y^{62} v_1^3-10706157143742069324300 x^{28} y^{63} v_1^3-4683941044594343779660 x^{27} y^{64} v_1^3-1945634933632411417158 x^{26} y^{65} v_1^3-766460698761489329871 x^{25} y^{66} v_1^3-285991739212574781750 x^{24} y^{67} v_1^3-100937584188272286000 x^{23} y^{68} v_1^3-33645457744342715700 x^{22} y^{69} v_1^3-10574062271535767235 x^{21} y^{70} v_1^3-3127423369493516940 x^{20} y^{71} v_1^3-868672829817431100 x^{19} y^{72} v_1^3-226067993795843850 x^{18} y^{73} v_1^3-54979215351891075 x^{17} y^{74} v_1^3-12458030271636726 x^{16} y^{75} v_1^3-2621365949752896 x^{15} y^{76} v_1^3-510212429469720 x^{14} y^{77} v_1^3-91446243902685 x^{13} y^{78} v_1^3-15013261314870 x^{12} y^{79} v_1^3-2243560298148 x^{11} y^{80} v_1^3-302849712022 x^{10} y^{81} v_1^3-36578261005 x^9 y^{82} v_1^3-3905708400 x^8 y^{83} v_1^3-362943120 x^7 y^{84} v_1^3-28735868 x^6 y^{85} v_1^3-1881201 x^5 y^{86} v_1^3-97350 x^4 y^{87} v_1^3-3700 x^3 y^{88} v_1^3-90 x^2 y^{89} v_1^3-x y^{90} v_1^3\right) d^{91}+\left(x^{120} y v_1^4+150 x^{119} y^2 v_1^4+9650 x^{118} y^3 v_1^4+382025 x^{117} y^4 v_1^4+10820586 x^{116} y^5 v_1^4+237933864 x^{115} y^6 v_1^4+4271856600 x^{114} y^7 v_1^4+64779664950 x^{113} y^8 v_1^4+849922943155 x^{112} y^9 v_1^4+9821986675358 x^{111} y^{10} v_1^4+101356334931306 x^{110} y^{11} v_1^4+944112998185175 x^{109} y^{12} v_1^4+8007470613301460 x^{108} y^{13} v_1^4+62282128589223840 x^{107} y^{14} v_1^4+446900549886216288 x^{106} y^{15} v_1^4+2973174173267819634 x^{105} y^{16} v_1^4+18418702050241365285 x^{104} y^{17} v_1^4+106645235395190398830 x^{103} y^{18} v_1^4+578998106814270645810 x^{102} y^{19} v_1^4+2956017768122273810571 x^{101} y^{20} v_1^4+14227611899431043141886 x^{100} y^{21} v_1^4+64704608636976356997000 x^{99} y^{22} v_1^4+278612079108564765447000 x^{98} y^{23} v_1^4+1137951981432518700357000 x^{97} y^{24} v_1^4+4416020148656934046715031 x^{96} y^{25} v_1^4+16307250799205388891595734 x^{95} y^{26} v_1^4+57382047864174666369764650 x^{94} y^{27} v_1^4+192650438272587264882105625 x^{93} y^{28} v_1^4+617833284394991912784832680 x^{92} y^{29} v_1^4+1894736805787157153965818752 x^{91} y^{30} v_1^4+5562063916528734874015571425 x^{90} y^{31} v_1^4+15643482109140847307276074245 x^{89} y^{32} v_1^4+42190314272850380375947181445 x^{88} y^{33} v_1^4+109199001354022988846321217090 x^{87} y^{34} v_1^4+271438398518221614315811490175 x^{86} y^{35} v_1^4+648437544475210315045043344346 x^{85} y^{36} v_1^4+1489655855228133863376694869890 x^{84} y^{37} v_1^4+3292926363907281552593672971320 x^{83} y^{38} v_1^4+7008026707960310500954821720605 x^{82} y^{39} v_1^4+14366459864856894942075451809583 x^{81} y^{40} v_1^4+28382524630338040919676739836423 x^{80} y^{41} v_1^4+54061959249147115996098039427050 x^{79} y^{42} v_1^4+99323143063394116957477609773915 x^{78} y^{43} v_1^4+176072853934832696406425093487660 x^{77} y^{44} v_1^4+301280226564668867396684839778364 x^{76} y^{45} v_1^4+497767340677707567235914476462208 x^{75} y^{46} v_1^4+794309595601192511849436161495475 x^{74} y^{47} v_1^4+1224560635180752902791183587273825 x^{73} y^{48} v_1^4+1824345443657923882972269971756625 x^{72} y^{49} v_1^4+2627057445228151852479563714018778 x^{71} y^{50} v_1^4+3657276056313510012905188928318001 x^{70} y^{51} v_1^4+4923256233586278673844242319825550 x^{69} y^{52} v_1^4+6409522269261453861726201550947510 x^{68} y^{53} v_1^4+8071250267032616893520658372577320 x^{67} y^{54} v_1^4+9832250326664167607719657914471855 x^{66} y^{55} v_1^4+11588009314448446716316668600052731 x^{65} y^{56} v_1^4+13214396587190401893137733950650035 x^{64} y^{57} v_1^4+14581403131004758184636860111557330 x^{63} y^{58} v_1^4+15569972834967602848391769643901325 x^{62} y^{59} v_1^4+16088971929537072870763437443081232 x^{61} y^{60} v_1^4+16088971929537072870763437443081232 x^{60} y^{61} v_1^4+15569972834967602848391769643901325 x^{59} y^{62} v_1^4+14581403131004758184636860111557330 x^{58} y^{63} v_1^4+13214396587190401893137733950650035 x^{57} y^{64} v_1^4+11588009314448446716316668600052731 x^{56} y^{65} v_1^4+9832250326664167607719657914471855 x^{55} y^{66} v_1^4+8071250267032616893520658372577320 x^{54} y^{67} v_1^4+6409522269261453861726201550947510 x^{53} y^{68} v_1^4+4923256233586278673844242319825550 x^{52} y^{69} v_1^4+3657276056313510012905188928318001 x^{51} y^{70} v_1^4+2627057445228151852479563714018778 x^{50} y^{71} v_1^4+1824345443657923882972269971756625 x^{49} y^{72} v_1^4+1224560635180752902791183587273825 x^{48} y^{73} v_1^4+794309595601192511849436161495475 x^{47} y^{74} v_1^4+497767340677707567235914476462208 x^{46} y^{75} v_1^4+301280226564668867396684839778364 x^{45} y^{76} v_1^4+176072853934832696406425093487660 x^{44} y^{77} v_1^4+99323143063394116957477609773915 x^{43} y^{78} v_1^4+54061959249147115996098039427050 x^{42} y^{79} v_1^4+28382524630338040919676739836423 x^{41} y^{80} v_1^4+14366459864856894942075451809583 x^{40} y^{81} v_1^4+7008026707960310500954821720605 x^{39} y^{82} v_1^4+3292926363907281552593672971320 x^{38} y^{83} v_1^4+1489655855228133863376694869890 x^{37} y^{84} v_1^4+648437544475210315045043344346 x^{36} y^{85} v_1^4+271438398518221614315811490175 x^{35} y^{86} v_1^4+109199001354022988846321217090 x^{34} y^{87} v_1^4+42190314272850380375947181445 x^{33} y^{88} v_1^4+15643482109140847307276074245 x^{32} y^{89} v_1^4+5562063916528734874015571425 x^{31} y^{90} v_1^4+1894736805787157153965818752 x^{30} y^{91} v_1^4+617833284394991912784832680 x^{29} y^{92} v_1^4+192650438272587264882105625 x^{28} y^{93} v_1^4+57382047864174666369764650 x^{27} y^{94} v_1^4+16307250799205388891595734 x^{26} y^{95} v_1^4+4416020148656934046715031 x^{25} y^{96} v_1^4+1137951981432518700357000 x^{24} y^{97} v_1^4+278612079108564765447000 x^{23} y^{98} v_1^4+64704608636976356997000 x^{22} y^{99} v_1^4+14227611899431043141886 x^{21} y^{100} v_1^4+2956017768122273810571 x^{20} y^{101} v_1^4+578998106814270645810 x^{19} y^{102} v_1^4+106645235395190398830 x^{18} y^{103} v_1^4+18418702050241365285 x^{17} y^{104} v_1^4+2973174173267819634 x^{16} y^{105} v_1^4+446900549886216288 x^{15} y^{106} v_1^4+62282128589223840 x^{14} y^{107} v_1^4+8007470613301460 x^{13} y^{108} v_1^4+944112998185175 x^{12} y^{109} v_1^4+101356334931306 x^{11} y^{110} v_1^4+9821986675358 x^{10} y^{111} v_1^4+849922943155 x^9 y^{112} v_1^4+64779664950 x^8 y^{113} v_1^4+4271856600 x^7 y^{114} v_1^4+237933864 x^6 y^{115} v_1^4+10820586 x^5 y^{116} v_1^4+382025 x^4 y^{117} v_1^4+9650 x^3 y^{118} v_1^4+150 x^2 y^{119} v_1^4+x y^{120} v_1^4\right) d^{121}+O[d]^{122}$
$(x+y) d+\left(-x^{36} y v_1-18 x^{35} y^2 v_1-210 x^{34} y^3 v_1-1785 x^{33} y^4 v_1-11781 x^{32} y^5 v_1-62832 x^{31} y^6 v_1-278256 x^{30} y^7 v_1-1043460 x^{29} y^8 v_1-3362260 x^{28} y^9 v_1-9414328 x^{27} y^{10} v_1-23107896 x^{26} y^{11} v_1-50067108 x^{25} y^{12} v_1-96282900 x^{24} y^{13} v_1-165056400 x^{23} y^{14} v_1-253086480 x^{22} y^{15} v_1-347993910 x^{21} y^{16} v_1-429874830 x^{20} y^{17} v_1-477638700 x^{19} y^{18} v_1-477638700 x^{18} y^{19} v_1-429874830 x^{17} y^{20} v_1-347993910 x^{16} y^{21} v_1-253086480 x^{15} y^{22} v_1-165056400 x^{14} y^{23} v_1-96282900 x^{13} y^{24} v_1-50067108 x^{12} y^{25} v_1-23107896 x^{11} y^{26} v_1-9414328 x^{10} y^{27} v_1-3362260 x^9 y^{28} v_1-1043460 x^8 y^{29} v_1-278256 x^7 y^{30} v_1-62832 x^6 y^{31} v_1-11781 x^5 y^{32} v_1-1785 x^4 y^{33} v_1-210 x^3 y^{34} v_1-18 x^2 y^{35} v_1-x y^{36} v_1\right) d^{37}+\left(x^{72} y v_1^2+54 x^{71} y^2 v_1^2+1488 x^{70} y^3 v_1^2+27825 x^{69} y^4 v_1^2+395766 x^{68} y^5 v_1^2+4548180 x^{67} y^6 v_1^2+43810836 x^{66} y^7 v_1^2+362482857 x^{65} y^8 v_1^2+2621294005 x^{64} y^9 v_1^2+16785695960 x^{63} y^{10} v_1^2+96159366576 x^{62} y^{11} v_1^2+496873461084 x^{61} y^{12} v_1^2+2331579446448 x^{60} y^{13} v_1^2+9992648398320 x^{59} y^{14} v_1^2+39304670119872 x^{58} y^{15} v_1^2+142479777178446 x^{57} y^{16} v_1^2+477726741590796 x^{56} y^{17} v_1^2+1486261451476732 x^{55} y^{18} v_1^2+4302336258229240 x^{54} y^{19} v_1^2+11616308327093778 x^{53} y^{20} v_1^2+29317349935421064 x^{52} y^{21} v_1^2+69295554645899904 x^{51} y^{22} v_1^2+153655360466834448 x^{50} y^{23} v_1^2+320115334402188000 x^{49} y^{24} v_1^2+627426055478355588 x^{48} y^{25} v_1^2+1158325025521610520 x^{47} y^{26} v_1^2+2016343562954440048 x^{46} y^{27} v_1^2+3312564424857085196 x^{45} y^{28} v_1^2+5140186176503417040 x^{44} y^{29} v_1^2+7538939725538623248 x^{43} y^{30} v_1^2+10457238974134282176 x^{42} y^{31} v_1^2+13725126153551257137 x^{41} y^{32} v_1^2+17052429463503078834 x^{40} y^{33} v_1^2+20061681721768328250 x^{39} y^{34} v_1^2+22354445347113280068 x^{38} y^{35} v_1^2+23596358977508462295 x^{37} y^{36} v_1^2+23596358977508462295 x^{36} y^{37} v_1^2+22354445347113280068 x^{35} y^{38} v_1^2+20061681721768328250 x^{34} y^{39} v_1^2+17052429463503078834 x^{33} y^{40} v_1^2+13725126153551257137 x^{32} y^{41} v_1^2+10457238974134282176 x^{31} y^{42} v_1^2+7538939725538623248 x^{30} y^{43} v_1^2+5140186176503417040 x^{29} y^{44} v_1^2+3312564424857085196 x^{28} y^{45} v_1^2+2016343562954440048 x^{27} y^{46} v_1^2+1158325025521610520 x^{26} y^{47} v_1^2+627426055478355588 x^{25} y^{48} v_1^2+320115334402188000 x^{24} y^{49} v_1^2+153655360466834448 x^{23} y^{50} v_1^2+69295554645899904 x^{22} y^{51} v_1^2+29317349935421064 x^{21} y^{52} v_1^2+11616308327093778 x^{20} y^{53} v_1^2+4302336258229240 x^{19} y^{54} v_1^2+1486261451476732 x^{18} y^{55} v_1^2+477726741590796 x^{17} y^{56} v_1^2+142479777178446 x^{16} y^{57} v_1^2+39304670119872 x^{15} y^{58} v_1^2+9992648398320 x^{14} y^{59} v_1^2+2331579446448 x^{13} y^{60} v_1^2+496873461084 x^{12} y^{61} v_1^2+96159366576 x^{11} y^{62} v_1^2+16785695960 x^{10} y^{63} v_1^2+2621294005 x^9 y^{64} v_1^2+362482857 x^8 y^{65} v_1^2+43810836 x^7 y^{66} v_1^2+4548180 x^6 y^{67} v_1^2+395766 x^5 y^{68} v_1^2+27825 x^4 y^{69} v_1^2+1488 x^3 y^{70} v_1^2+54 x^2 y^{71} v_1^2+x y^{72} v_1^2\right) d^{73}+\left(-x^{108} y v_1^3-108 x^{107} y^2 v_1^3-5340 x^{106} y^3 v_1^3-169335 x^{105} y^4 v_1^3-3951801 x^{104} y^5 v_1^3-73046064 x^{103} y^6 v_1^3-1118631492 x^{102} y^7 v_1^3-14625034380 x^{101} y^8 v_1^3-166746679825 x^{100} y^9 v_1^3-1684252494210 x^{99} y^{10} v_1^3-15254431814466 x^{98} y^{11} v_1^3-125074733279223 x^{97} y^{12} v_1^3-935581512376035 x^{96} y^{13} v_1^3-6425408733262560 x^{95} y^{14} v_1^3-40733559980782752 x^{94} y^{15} v_1^3-239452144664277114 x^{93} y^{16} v_1^3-1310421812257930302 x^{92} y^{17} v_1^3-6699197746325342720 x^{91} y^{18} v_1^3-32089933647605923320 x^{90} y^{19} v_1^3-144416317722553748718 x^{89} y^{20} v_1^3-612079425793139403726 x^{88} y^{21} v_1^3-2448386998727203514808 x^{87} y^{22} v_1^3-9261443607067714912200 x^{86} y^{23} v_1^3-33187159707327047290050 x^{85} y^{24} v_1^3-112836970430967439141758 x^{84} y^{25} v_1^3-364551370486612632683892 x^{83} y^{26} v_1^3-1120659933024631417875716 x^{82} y^{27} v_1^3-3281935973565131152292650 x^{81} y^{28} v_1^3-9166791824971542825337890 x^{80} y^{29} v_1^3-24444785738863839739524288 x^{79} y^{30} v_1^3-62294787017569404438231168 x^{78} y^{31} v_1^3-151843557080451576869445609 x^{77} y^{32} v_1^3-354301650240149809531785255 x^{76} y^{33} v_1^3-791968414716134237192318820 x^{75} y^{34} v_1^3-1697075196746161569668248968 x^{74} y^{35} v_1^3-3488432372463468870715418507 x^{73} y^{36} v_1^3-6882582812510770533244287998 x^{72} y^{37} v_1^3-13040683246059063199576141536 x^{71} y^{38} v_1^3-23740731057758950623560790948 x^{70} y^{39} v_1^3-41546279368130593054734460389 x^{69} y^{40} v_1^3-69919348218627831538348229013 x^{68} y^{41} v_1^3-113202754269188013845745059814 x^{67} y^{42} v_1^3-176385686892087705485180815794 x^{66} y^{43} v_1^3-264578530343271744404208924477 x^{65} y^{44} v_1^3-382168988276927306341524167536 x^{64} y^{45} v_1^3-531713374996002161077949742352 x^{63} y^{46} v_1^3-712722183506437817511762779088 x^{62} y^{47} v_1^3-920599487029776273550632763416 x^{61} y^{48} v_1^3-1146052422629225271585000503148 x^{60} y^{49} v_1^3-1375262907155223978464572102488 x^{59} y^{50} v_1^3-1590990422003171533590792930216 x^{58} y^{51} v_1^3-1774566239926643676361380094140 x^{57} y^{52} v_1^3-1908495767468288624445145323864 x^{56} y^{53} v_1^3-1979180795893043861896084322792 x^{55} y^{54} v_1^3-1979180795893043861896084322792 x^{54} y^{55} v_1^3-1908495767468288624445145323864 x^{53} y^{56} v_1^3-1774566239926643676361380094140 x^{52} y^{57} v_1^3-1590990422003171533590792930216 x^{51} y^{58} v_1^3-1375262907155223978464572102488 x^{50} y^{59} v_1^3-1146052422629225271585000503148 x^{49} y^{60} v_1^3-920599487029776273550632763416 x^{48} y^{61} v_1^3-712722183506437817511762779088 x^{47} y^{62} v_1^3-531713374996002161077949742352 x^{46} y^{63} v_1^3-382168988276927306341524167536 x^{45} y^{64} v_1^3-264578530343271744404208924477 x^{44} y^{65} v_1^3-176385686892087705485180815794 x^{43} y^{66} v_1^3-113202754269188013845745059814 x^{42} y^{67} v_1^3-69919348218627831538348229013 x^{41} y^{68} v_1^3-41546279368130593054734460389 x^{40} y^{69} v_1^3-23740731057758950623560790948 x^{39} y^{70} v_1^3-13040683246059063199576141536 x^{38} y^{71} v_1^3-6882582812510770533244287998 x^{37} y^{72} v_1^3-3488432372463468870715418507 x^{36} y^{73} v_1^3-1697075196746161569668248968 x^{35} y^{74} v_1^3-791968414716134237192318820 x^{34} y^{75} v_1^3-354301650240149809531785255 x^{33} y^{76} v_1^3-151843557080451576869445609 x^{32} y^{77} v_1^3-62294787017569404438231168 x^{31} y^{78} v_1^3-24444785738863839739524288 x^{30} y^{79} v_1^3-9166791824971542825337890 x^{29} y^{80} v_1^3-3281935973565131152292650 x^{28} y^{81} v_1^3-1120659933024631417875716 x^{27} y^{82} v_1^3-364551370486612632683892 x^{26} y^{83} v_1^3-112836970430967439141758 x^{25} y^{84} v_1^3-33187159707327047290050 x^{24} y^{85} v_1^3-9261443607067714912200 x^{23} y^{86} v_1^3-2448386998727203514808 x^{22} y^{87} v_1^3-612079425793139403726 x^{21} y^{88} v_1^3-144416317722553748718 x^{20} y^{89} v_1^3-32089933647605923320 x^{19} y^{90} v_1^3-6699197746325342720 x^{18} y^{91} v_1^3-1310421812257930302 x^{17} y^{92} v_1^3-239452144664277114 x^{16} y^{93} v_1^3-40733559980782752 x^{15} y^{94} v_1^3-6425408733262560 x^{14} y^{95} v_1^3-935581512376035 x^{13} y^{96} v_1^3-125074733279223 x^{12} y^{97} v_1^3-15254431814466 x^{11} y^{98} v_1^3-1684252494210 x^{10} y^{99} v_1^3-166746679825 x^9 y^{100} v_1^3-14625034380 x^8 y^{101} v_1^3-1118631492 x^7 y^{102} v_1^3-73046064 x^6 y^{103} v_1^3-3951801 x^5 y^{104} v_1^3-169335 x^4 y^{105} v_1^3-5340 x^3 y^{106} v_1^3-108 x^2 y^{107} v_1^3-x y^{108} v_1^3\right) d^{109}+\left(x^{144} y v_1^4+180 x^{143} y^2 v_1^4+13920 x^{142} y^3 v_1^4+663495 x^{141} y^4 v_1^4+22662360 x^{140} y^5 v_1^4+601834464 x^{139} y^6 v_1^4+13069344420 x^{138} y^7 v_1^4+240071225625 x^{137} y^8 v_1^4+3821164225450 x^{136} y^9 v_1^4+53652085960330 x^{135} y^{10} v_1^4+673711850418516 x^{134} y^{11} v_1^4+7648190396285985 x^{133} y^{12} v_1^4+79182452489763420 x^{132} y^{13} v_1^4+753002817922460520 x^{131} y^{14} v_1^4+6616958169836937960 x^{130} y^{15} v_1^4+54002237274589398039 x^{129} y^{16} v_1^4+411092104660612774245 x^{128} y^{17} v_1^4+2930020830888460626240 x^{127} y^{18} v_1^4+19616966013796790109240 x^{126} y^{19} v_1^4+123731302204642331436930 x^{125} y^{20} v_1^4+737107925881997493194976 x^{124} y^{21} v_1^4+4157056696515440346977400 x^{123} y^{22} v_1^4+22240477690189640005269600 x^{122} y^{23} v_1^4+113088948751504663740743850 x^{121} y^{24} v_1^4+547463348927713539944341992 x^{120} y^{25} v_1^4+2527118469498395258529646932 x^{119} y^{26} v_1^4+11139161321796322252344838120 x^{118} y^{27} v_1^4+46946890363543780337462681870 x^{117} y^{28} v_1^4+189415586534398154283623054400 x^{116} y^{29} v_1^4+732431379385411727069748667968 x^{115} y^{30} v_1^4+2717146444119996556921247805888 x^{114} y^{31} v_1^4+9679986050734568185608814754085 x^{113} y^{32} v_1^4+33146973202650428179318503518940 x^{112} y^{33} v_1^4+109190821341851420724933439204740 x^{111} y^{34} v_1^4+346292587616496966174930004012572 x^{110} y^{35} v_1^4+1058119728371668749003379061012477 x^{109} y^{36} v_1^4+3117170406704755582158866153757187 x^{108} y^{37} v_1^4+8859339459738867187304187592083012 x^{107} y^{38} v_1^4+24306416617450513888221086954454300 x^{106} y^{39} v_1^4+64412045582523229934378935163750133 x^{105} y^{40} v_1^4+164957747630688197728070274742765203 x^{104} y^{41} v_1^4+408466916859696473086092621288788238 x^{103} y^{42} v_1^4+978420930724029606689276182511547048 x^{102} y^{43} v_1^4+2268157876711508067869611915619774175 x^{101} y^{44} v_1^4+5090754727677039718145768608201250187 x^{100} y^{45} v_1^4+11066858635359113513710353341981417052 x^{99} y^{46} v_1^4+23311043370180741758721324195064127202 x^{98} y^{47} v_1^4+47593381134718501453832279304423735252 x^{97} y^{48} v_1^4+94215469922944149997015454447857120185 x^{96} y^{49} v_1^4+180893703627315675149491854687954011256 x^{95} y^{50} v_1^4+336958861288931385516918901856008050336 x^{94} y^{51} v_1^4+609117943335326821437923442127275539388 x^{93} y^{52} v_1^4+1068829600591239057915545437056142444458 x^{92} y^{53} v_1^4+1820968951134625116783962776756035717280 x^{91} y^{54} v_1^4+3012875902947378716379061783530332397064 x^{90} y^{55} v_1^4+4842121988788211561597350526042384299524 x^{89} y^{56} v_1^4+7560506264970194818335611124439712006802 x^{88} y^{57} v_1^4+11471112955338872214411288593606282396952 x^{87} y^{58} v_1^4+16915030969417328711236112770913383335480 x^{86} y^{59} v_1^4+24244877723977556895459584406430458576516 x^{85} y^{60} v_1^4+33783846009741785278720859870661943572426 x^{84} y^{61} v_1^4+45771662336491914988897513832581736523864 x^{83} y^{62} v_1^4+60302348793052807245166317997830896071204 x^{82} y^{63} v_1^4+77262384391481076835300733053253355605505 x^{81} y^{64} v_1^4+96280817472725608650690125875756493052804 x^{80} y^{65} v_1^4+116704021179237718394647603953016899258114 x^{79} y^{66} v_1^4+137606233927871080695151088641382993104176 x^{78} y^{67} v_1^4+157842444799686675748636238767043161171581 x^{77} y^{68} v_1^4+176143018109836652666195597119009871603865 x^{76} y^{69} v_1^4+191240991090703150383992717945755358308296 x^{75} y^{70} v_1^4+202015131433853560136321413089451159933236 x^{74} y^{71} v_1^4+207626662862576852951190640668633666296829 x^{73} y^{72} v_1^4+207626662862576852951190640668633666296829 x^{72} y^{73} v_1^4+202015131433853560136321413089451159933236 x^{71} y^{74} v_1^4+191240991090703150383992717945755358308296 x^{70} y^{75} v_1^4+176143018109836652666195597119009871603865 x^{69} y^{76} v_1^4+157842444799686675748636238767043161171581 x^{68} y^{77} v_1^4+137606233927871080695151088641382993104176 x^{67} y^{78} v_1^4+116704021179237718394647603953016899258114 x^{66} y^{79} v_1^4+96280817472725608650690125875756493052804 x^{65} y^{80} v_1^4+77262384391481076835300733053253355605505 x^{64} y^{81} v_1^4+60302348793052807245166317997830896071204 x^{63} y^{82} v_1^4+45771662336491914988897513832581736523864 x^{62} y^{83} v_1^4+33783846009741785278720859870661943572426 x^{61} y^{84} v_1^4+24244877723977556895459584406430458576516 x^{60} y^{85} v_1^4+16915030969417328711236112770913383335480 x^{59} y^{86} v_1^4+11471112955338872214411288593606282396952 x^{58} y^{87} v_1^4+7560506264970194818335611124439712006802 x^{57} y^{88} v_1^4+4842121988788211561597350526042384299524 x^{56} y^{89} v_1^4+3012875902947378716379061783530332397064 x^{55} y^{90} v_1^4+1820968951134625116783962776756035717280 x^{54} y^{91} v_1^4+1068829600591239057915545437056142444458 x^{53} y^{92} v_1^4+609117943335326821437923442127275539388 x^{52} y^{93} v_1^4+336958861288931385516918901856008050336 x^{51} y^{94} v_1^4+180893703627315675149491854687954011256 x^{50} y^{95} v_1^4+94215469922944149997015454447857120185 x^{49} y^{96} v_1^4+47593381134718501453832279304423735252 x^{48} y^{97} v_1^4+23311043370180741758721324195064127202 x^{47} y^{98} v_1^4+11066858635359113513710353341981417052 x^{46} y^{99} v_1^4+5090754727677039718145768608201250187 x^{45} y^{100} v_1^4+2268157876711508067869611915619774175 x^{44} y^{101} v_1^4+978420930724029606689276182511547048 x^{43} y^{102} v_1^4+408466916859696473086092621288788238 x^{42} y^{103} v_1^4+164957747630688197728070274742765203 x^{41} y^{104} v_1^4+64412045582523229934378935163750133 x^{40} y^{105} v_1^4+24306416617450513888221086954454300 x^{39} y^{106} v_1^4+8859339459738867187304187592083012 x^{38} y^{107} v_1^4+3117170406704755582158866153757187 x^{37} y^{108} v_1^4+1058119728371668749003379061012477 x^{36} y^{109} v_1^4+346292587616496966174930004012572 x^{35} y^{110} v_1^4+109190821341851420724933439204740 x^{34} y^{111} v_1^4+33146973202650428179318503518940 x^{33} y^{112} v_1^4+9679986050734568185608814754085 x^{32} y^{113} v_1^4+2717146444119996556921247805888 x^{31} y^{114} v_1^4+732431379385411727069748667968 x^{30} y^{115} v_1^4+189415586534398154283623054400 x^{29} y^{116} v_1^4+46946890363543780337462681870 x^{28} y^{117} v_1^4+11139161321796322252344838120 x^{27} y^{118} v_1^4+2527118469498395258529646932 x^{26} y^{119} v_1^4+547463348927713539944341992 x^{25} y^{120} v_1^4+113088948751504663740743850 x^{24} y^{121} v_1^4+22240477690189640005269600 x^{23} y^{122} v_1^4+4157056696515440346977400 x^{22} y^{123} v_1^4+737107925881997493194976 x^{21} y^{124} v_1^4+123731302204642331436930 x^{20} y^{125} v_1^4+19616966013796790109240 x^{19} y^{126} v_1^4+2930020830888460626240 x^{18} y^{127} v_1^4+411092104660612774245 x^{17} y^{128} v_1^4+54002237274589398039 x^{16} y^{129} v_1^4+6616958169836937960 x^{15} y^{130} v_1^4+753002817922460520 x^{14} y^{131} v_1^4+79182452489763420 x^{13} y^{132} v_1^4+7648190396285985 x^{12} y^{133} v_1^4+673711850418516 x^{11} y^{134} v_1^4+53652085960330 x^{10} y^{135} v_1^4+3821164225450 x^9 y^{136} v_1^4+240071225625 x^8 y^{137} v_1^4+13069344420 x^7 y^{138} v_1^4+601834464 x^6 y^{139} v_1^4+22662360 x^5 y^{140} v_1^4+663495 x^4 y^{141} v_1^4+13920 x^3 y^{142} v_1^4+180 x^2 y^{143} v_1^4+x y^{144} v_1^4\right) d^{145}+O[d]^{146}$
$(x+y) d+\left(-x^{40} y v_1-20 x^{39} y^2 v_1-260 x^{38} y^3 v_1-2470 x^{37} y^4 v_1-18278 x^{36} y^5 v_1-109668 x^{35} y^6 v_1-548340 x^{34} y^7 v_1-2330445 x^{33} y^8 v_1-8544965 x^{32} y^9 v_1-27343888 x^{31} y^{10} v_1-77060048 x^{30} y^{11} v_1-192650120 x^{29} y^{12} v_1-429757960 x^{28} y^{13} v_1-859515920 x^{27} y^{14} v_1-1547128656 x^{26} y^{15} v_1-2514084066 x^{25} y^{16} v_1-3697182450 x^{24} y^{17} v_1-4929576600 x^{23} y^{18} v_1-5967382200 x^{22} y^{19} v_1-6564120420 x^{21} y^{20} v_1-6564120420 x^{20} y^{21} v_1-5967382200 x^{19} y^{22} v_1-4929576600 x^{18} y^{23} v_1-3697182450 x^{17} y^{24} v_1-2514084066 x^{16} y^{25} v_1-1547128656 x^{15} y^{26} v_1-859515920 x^{14} y^{27} v_1-429757960 x^{13} y^{28} v_1-192650120 x^{12} y^{29} v_1-77060048 x^{11} y^{30} v_1-27343888 x^{10} y^{31} v_1-8544965 x^9 y^{32} v_1-2330445 x^8 y^{33} v_1-548340 x^7 y^{34} v_1-109668 x^6 y^{35} v_1-18278 x^5 y^{36} v_1-2470 x^4 y^{37} v_1-260 x^3 y^{38} v_1-20 x^2 y^{39} v_1-x y^{40} v_1\right) d^{41}+\left(x^{80} y v_1^2+60 x^{79} y^2 v_1^2+1840 x^{78} y^3 v_1^2+38350 x^{77} y^4 v_1^2+608868 x^{76} y^5 v_1^2+7821996 x^{75} y^6 v_1^2+84355440 x^{74} y^7 v_1^2+782618265 x^{73} y^8 v_1^2+6356448670 x^{72} y^9 v_1^2+45793774312 x^{71} y^{10} v_1^2+295655057880 x^{70} y^{11} v_1^2+1724847154420 x^{69} y^{12} v_1^2+9155387731420 x^{68} y^{13} v_1^2+44469885639960 x^{67} y^{14} v_1^2+198633702987144 x^{66} y^{15} v_1^2+819366538906035 x^{65} y^{16} v_1^2+3132875757705525 x^{64} y^{17} v_1^2+11139118734751800 x^{63} y^{18} v_1^2+36934978614190800 x^{62} y^{19} v_1^2+114498440268111900 x^{61} y^{20} v_1^2+332590714009588320 x^{60} y^{21} v_1^2+907065589629895800 x^{59} y^{22} v_1^2+2326820430501918000 x^{58} y^{23} v_1^2+5623149377410150950 x^{57} y^{24} v_1^2+12820780583009228232 x^{56} y^{25} v_1^2+27613988949567004848 x^{55} y^{26} v_1^2+56250718231458970240 x^{54} y^{27} v_1^2+108483528018243486280 x^{53} y^{28} v_1^2+198262999481810056080 x^{52} y^{29} v_1^2+343655865768547823920 x^{51} y^{30} v_1^2+565369327554735054208 x^{50} y^{31} v_1^2+883389574304282067165 x^{49} y^{32} v_1^2+1311699670936663581690 x^{48} y^{33} v_1^2+1851811300145878546020 x^{47} y^{34} v_1^2+2486718031624465585752 x^{46} y^{35} v_1^2+3177473040409039377850 x^{45} y^{36} v_1^2+3864494238335318164720 x^{44} y^{37} v_1^2+4474677539125105243620 x^{43} y^{38} v_1^2+4933618825189218601960 x^{42} y^{39} v_1^2+5180299766448679532059 x^{41} y^{40} v_1^2+5180299766448679532059 x^{40} y^{41} v_1^2+4933618825189218601960 x^{39} y^{42} v_1^2+4474677539125105243620 x^{38} y^{43} v_1^2+3864494238335318164720 x^{37} y^{44} v_1^2+3177473040409039377850 x^{36} y^{45} v_1^2+2486718031624465585752 x^{35} y^{46} v_1^2+1851811300145878546020 x^{34} y^{47} v_1^2+1311699670936663581690 x^{33} y^{48} v_1^2+883389574304282067165 x^{32} y^{49} v_1^2+565369327554735054208 x^{31} y^{50} v_1^2+343655865768547823920 x^{30} y^{51} v_1^2+198262999481810056080 x^{29} y^{52} v_1^2+108483528018243486280 x^{28} y^{53} v_1^2+56250718231458970240 x^{27} y^{54} v_1^2+27613988949567004848 x^{26} y^{55} v_1^2+12820780583009228232 x^{25} y^{56} v_1^2+5623149377410150950 x^{24} y^{57} v_1^2+2326820430501918000 x^{23} y^{58} v_1^2+907065589629895800 x^{22} y^{59} v_1^2+332590714009588320 x^{21} y^{60} v_1^2+114498440268111900 x^{20} y^{61} v_1^2+36934978614190800 x^{19} y^{62} v_1^2+11139118734751800 x^{18} y^{63} v_1^2+3132875757705525 x^{17} y^{64} v_1^2+819366538906035 x^{16} y^{65} v_1^2+198633702987144 x^{15} y^{66} v_1^2+44469885639960 x^{14} y^{67} v_1^2+9155387731420 x^{13} y^{68} v_1^2+1724847154420 x^{12} y^{69} v_1^2+295655057880 x^{11} y^{70} v_1^2+45793774312 x^{10} y^{71} v_1^2+6356448670 x^9 y^{72} v_1^2+782618265 x^8 y^{73} v_1^2+84355440 x^7 y^{74} v_1^2+7821996 x^6 y^{75} v_1^2+608868 x^5 y^{76} v_1^2+38350 x^4 y^{77} v_1^2+1840 x^3 y^{78} v_1^2+60 x^2 y^{79} v_1^2+x y^{80} v_1^2\right) d^{81}+\left(-x^{120} y v_1^3-120 x^{119} y^2 v_1^3-6600 x^{118} y^3 v_1^3-233050 x^{117} y^4 v_1^3-6062238 x^{116} y^5 v_1^3-125025264 x^{115} y^6 v_1^3-2138341920 x^{114} y^7 v_1^3-31253990625 x^{113} y^8 v_1^3-398767664295 x^{112} y^9 v_1^3-4511991614416 x^{111} y^{10} v_1^3-45825752257896 x^{110} y^{11} v_1^3-421794242851800 x^{109} y^{12} v_1^3-3545737885488820 x^{108} y^{13} v_1^3-27397305002268000 x^{107} y^{14} v_1^3-195632742719165544 x^{106} y^{15} v_1^3-1296886287053377764 x^{105} y^{16} v_1^3-8013312884028568185 x^{104} y^{17} v_1^3-46310280226455367980 x^{103} y^{18} v_1^3-251087401469398554060 x^{102} y^{19} v_1^3-1280660245934200737606 x^{101} y^{20} v_1^3-6159698535445165516806 x^{100} y^{21} v_1^3-27999536772158564063100 x^{99} y^{22} v_1^3-120522072057113016798300 x^{98} y^{23} v_1^3-492137417382588895410675 x^{97} y^{24} v_1^3-1909506000225027923421651 x^{96} y^{25} v_1^3-7050511307127514207330944 x^{95} y^{26} v_1^3-24807410849870596262542080 x^{94} y^{27} v_1^3-83282130622379305696306120 x^{93} y^{28} v_1^3-267077375776146910422348120 x^{92} y^{29} v_1^3-819037629369382960509691488 x^{91} y^{30} v_1^3-2404272316098806567844148576 x^{90} y^{31} v_1^3-6762016772417467776343735035 x^{89} y^{32} v_1^3-18236955637310417363772442845 x^{88} y^{33} v_1^3-47201534089555909793289574560 x^{87} y^{34} v_1^3-117329530080757007396356813944 x^{86} y^{35} v_1^3-280287213925948113633669544494 x^{85} y^{36} v_1^3-643903062883564229115369820990 x^{84} y^{37} v_1^3-1423364669796240571906449058440 x^{83} y^{38} v_1^3-3029211994499976965400379418640 x^{82} y^{39} v_1^3-6209884593905252545519457340271 x^{81} y^{40} v_1^3-12268308593139457234426144033570 x^{80} y^{41} v_1^3-23368206849008775462191397713520 x^{79} y^{42} v_1^3-42932287006141962690593021973000 x^{78} y^{43} v_1^3-76107236060207064462568402568140 x^{77} y^{44} v_1^3-130227937261754005565248305928824 x^{76} y^{45} v_1^3-215159200695819422878556448288288 x^{75} y^{46} v_1^3-343339150048372166957416794183720 x^{74} y^{47} v_1^3-529314522992552123730287424566955 x^{73} y^{48} v_1^3-788570615887746757580649687419295 x^{72} y^{49} v_1^3-1135541686878920700243681131651908 x^{71} y^{50} v_1^3-1580852152321978356204947155457436 x^{70} y^{51} v_1^3-2128070205049015280967281906747790 x^{69} y^{52} v_1^3-2770506493365807622898271773900290 x^{68} y^{53} v_1^3-3488785954608851035097118760071780 x^{67} y^{54} v_1^3-4249975617432627965780358195858156 x^{66} y^{55} v_1^3-5008899834831324351644758304397639 x^{65} y^{56} v_1^3-5711903320421691286371315388343130 x^{64} y^{57} v_1^3-6302789870810144432462846646069840 x^{63} y^{58} v_1^3-6730097658661680543394316043676080 x^{62} y^{59} v_1^3-6954434247283736855932029353389776 x^{61} y^{60} v_1^3-6954434247283736855932029353389776 x^{60} y^{61} v_1^3-6730097658661680543394316043676080 x^{59} y^{62} v_1^3-6302789870810144432462846646069840 x^{58} y^{63} v_1^3-5711903320421691286371315388343130 x^{57} y^{64} v_1^3-5008899834831324351644758304397639 x^{56} y^{65} v_1^3-4249975617432627965780358195858156 x^{55} y^{66} v_1^3-3488785954608851035097118760071780 x^{54} y^{67} v_1^3-2770506493365807622898271773900290 x^{53} y^{68} v_1^3-2128070205049015280967281906747790 x^{52} y^{69} v_1^3-1580852152321978356204947155457436 x^{51} y^{70} v_1^3-1135541686878920700243681131651908 x^{50} y^{71} v_1^3-788570615887746757580649687419295 x^{49} y^{72} v_1^3-529314522992552123730287424566955 x^{48} y^{73} v_1^3-343339150048372166957416794183720 x^{47} y^{74} v_1^3-215159200695819422878556448288288 x^{46} y^{75} v_1^3-130227937261754005565248305928824 x^{45} y^{76} v_1^3-76107236060207064462568402568140 x^{44} y^{77} v_1^3-42932287006141962690593021973000 x^{43} y^{78} v_1^3-23368206849008775462191397713520 x^{42} y^{79} v_1^3-12268308593139457234426144033570 x^{41} y^{80} v_1^3-6209884593905252545519457340271 x^{40} y^{81} v_1^3-3029211994499976965400379418640 x^{39} y^{82} v_1^3-1423364669796240571906449058440 x^{38} y^{83} v_1^3-643903062883564229115369820990 x^{37} y^{84} v_1^3-280287213925948113633669544494 x^{36} y^{85} v_1^3-117329530080757007396356813944 x^{35} y^{86} v_1^3-47201534089555909793289574560 x^{34} y^{87} v_1^3-18236955637310417363772442845 x^{33} y^{88} v_1^3-6762016772417467776343735035 x^{32} y^{89} v_1^3-2404272316098806567844148576 x^{31} y^{90} v_1^3-819037629369382960509691488 x^{30} y^{91} v_1^3-267077375776146910422348120 x^{29} y^{92} v_1^3-83282130622379305696306120 x^{28} y^{93} v_1^3-24807410849870596262542080 x^{27} y^{94} v_1^3-7050511307127514207330944 x^{26} y^{95} v_1^3-1909506000225027923421651 x^{25} y^{96} v_1^3-492137417382588895410675 x^{24} y^{97} v_1^3-120522072057113016798300 x^{23} y^{98} v_1^3-27999536772158564063100 x^{22} y^{99} v_1^3-6159698535445165516806 x^{21} y^{100} v_1^3-1280660245934200737606 x^{20} y^{101} v_1^3-251087401469398554060 x^{19} y^{102} v_1^3-46310280226455367980 x^{18} y^{103} v_1^3-8013312884028568185 x^{17} y^{104} v_1^3-1296886287053377764 x^{16} y^{105} v_1^3-195632742719165544 x^{15} y^{106} v_1^3-27397305002268000 x^{14} y^{107} v_1^3-3545737885488820 x^{13} y^{108} v_1^3-421794242851800 x^{12} y^{109} v_1^3-45825752257896 x^{11} y^{110} v_1^3-4511991614416 x^{10} y^{111} v_1^3-398767664295 x^9 y^{112} v_1^3-31253990625 x^8 y^{113} v_1^3-2138341920 x^7 y^{114} v_1^3-125025264 x^6 y^{115} v_1^3-6062238 x^5 y^{116} v_1^3-233050 x^4 y^{117} v_1^3-6600 x^3 y^{118} v_1^3-120 x^2 y^{119} v_1^3-x y^{120} v_1^3\right) d^{121}+\left(x^{160} y v_1^4+200 x^{159} y^2 v_1^4+17200 x^{158} y^3 v_1^4+912450 x^{157} y^4 v_1^4+34713168 x^{156} y^5 v_1^4+1027567632 x^{155} y^6 v_1^4+24891625200 x^{154} y^7 v_1^4+510417775725 x^{153} y^8 v_1^4+9075869851620 x^{152} y^9 v_1^4+142465213359040 x^{151} y^{10} v_1^4+2001484590186536 x^{150} y^{11} v_1^4+25440351620183500 x^{149} y^{12} v_1^4+295131306455284320 x^{148} y^{13} v_1^4+3147356830386702240 x^{147} y^{14} v_1^4+31039729680508847496 x^{146} y^{15} v_1^4+284534419621696611165 x^{145} y^{16} v_1^4+2434924539069087898710 x^{144} y^{17} v_1^4+19525706592779158557660 x^{143} y^{18} v_1^4+147207721231333591909080 x^{142} y^{19} v_1^4+1046455480988402703292074 x^{141} y^{20} v_1^4+7032360785171863316192160 x^{140} y^{21} v_1^4+44779386351502197848922300 x^{139} y^{22} v_1^4+270743770022439960886372200 x^{138} y^{23} v_1^4+1557268815046412363992050825 x^{137} y^{24} v_1^4+8535742612454564782599860172 x^{136} y^{25} v_1^4+44655550330300235607806599536 x^{135} y^{26} v_1^4+223302559062351048635295539760 x^{134} y^{27} v_1^4+1068745529071873826346039246400 x^{133} y^{28} v_1^4+4901755193464025074635360960920 x^{132} y^{29} v_1^4+21568541888871079711356097919536 x^{131} y^{30} v_1^4+91146887738190661457782322453712 x^{130} y^{31} v_1^4+370290993453171979640017028703240 x^{129} y^{32} v_1^4+1447519393181673230828339430100965 x^{128} y^{33} v_1^4+5449531975865094660204718202895840 x^{127} y^{34} v_1^4+19774133356240567095464516693035992 x^{126} y^{35} v_1^4+69209747034055910782239442095170466 x^{125} y^{36} v_1^4+233817356855954474044767771096748240 x^{124} y^{37} v_1^4+762984377315679132597182527922657960 x^{123} y^{38} v_1^4+2406338373053751764321925680750878360 x^{122} y^{39} v_1^4+7339338247698536786434418845747519269 x^{121} y^{40} v_1^4+21660010511516469996983202239203785559 x^{120} y^{41} v_1^4+61885767686825334714441754303408529400 x^{119} y^{42} v_1^4+171265306995826885933092196688501391240 x^{118} y^{43} v_1^4+459302490323317254300357171869383554220 x^{117} y^{44} v_1^4+1194186605068562122934934212108702563866 x^{116} y^{45} v_1^4+3011427306201661701481430891700117727176 x^{115} y^{46} v_1^4+7368386305321939317954391479661457177760 x^{114} y^{47} v_1^4+17499918004454128872693803494478306802075 x^{113} y^{48} v_1^4+40356954553944423288040630945394931657945 x^{112} y^{49} v_1^4+90399579336377195044131713560472538997684 x^{111} y^{50} v_1^4+196752027195320165065088556297331876232784 x^{110} y^{51} v_1^4+416206213502785938840548765345467938233130 x^{109} y^{52} v_1^4+855971272049820971358634328118657298363780 x^{108} y^{53} v_1^4+1711942547588427897326119684242957585821700 x^{107} y^{54} v_1^4+3330506415012917163139806207280082312953536 x^{106} y^{55} v_1^4+6304172861997635893631671445120079678353481 x^{105} y^{56} v_1^4+11612950014654916808690557348907249171193135 x^{104} y^{57} v_1^4+20823220722235744148462163930769984143190440 x^{103} y^{58} v_1^4+36352402284531481511061648592723893024111900 x^{102} y^{59} v_1^4+61799083890657952816088112614980149516838104 x^{101} y^{60} v_1^4+102323073334109405303427507797377205532506994 x^{100} y^{61} v_1^4+165037215061745267502889661347977800506563480 x^{99} y^{62} v_1^4+259344195103331067375307041564357426841427580 x^{98} y^{63} v_1^4+397120798757687600238676049258956170932306715 x^{97} y^{64} v_1^4+592626422766481164805659623561377642267150191 x^{96} y^{65} v_1^4+862002069482768033513796435327456888150372396 x^{95} y^{66} v_1^4+1222241740314876296150741131362429813156921600 x^{94} y^{67} v_1^4+1689569464555687739373333009507906601010248450 x^{93} y^{68} v_1^4+2277245800055446327509030634648417114721191800 x^{92} y^{69} v_1^4+2992951622931596025098889118588331501376794844 x^{91} y^{70} v_1^4+3836036587138814953087815336459741106750051984 x^{90} y^{71} v_1^4+4795045733924307258970316522208748823319798555 x^{89} y^{72} v_1^4+5846014661908246375434481288170634577573466315 x^{88} y^{73} v_1^4+6952017435783122790952511308325053562935941360 x^{87} y^{74} v_1^4+8064340225508637541951829393195085329238073776 x^{86} y^{75} v_1^4+9125437623602009418957232028606548172355775476 x^{85} y^{76} v_1^4+10073535039041255026547119151557539656562693250 x^{84} y^{77} v_1^4+10848422349736778421288297507493948098921989280 x^{83} y^{78} v_1^4+11397709557318409328050660401611287811337628200 x^{82} y^{79} v_1^4+11682652296251378724618225688632414897376198369 x^{81} y^{80} v_1^4+11682652296251378724618225688632414897376198369 x^{80} y^{81} v_1^4+11397709557318409328050660401611287811337628200 x^{79} y^{82} v_1^4+10848422349736778421288297507493948098921989280 x^{78} y^{83} v_1^4+10073535039041255026547119151557539656562693250 x^{77} y^{84} v_1^4+9125437623602009418957232028606548172355775476 x^{76} y^{85} v_1^4+8064340225508637541951829393195085329238073776 x^{75} y^{86} v_1^4+6952017435783122790952511308325053562935941360 x^{74} y^{87} v_1^4+5846014661908246375434481288170634577573466315 x^{73} y^{88} v_1^4+4795045733924307258970316522208748823319798555 x^{72} y^{89} v_1^4+3836036587138814953087815336459741106750051984 x^{71} y^{90} v_1^4+2992951622931596025098889118588331501376794844 x^{70} y^{91} v_1^4+2277245800055446327509030634648417114721191800 x^{69} y^{92} v_1^4+1689569464555687739373333009507906601010248450 x^{68} y^{93} v_1^4+1222241740314876296150741131362429813156921600 x^{67} y^{94} v_1^4+862002069482768033513796435327456888150372396 x^{66} y^{95} v_1^4+592626422766481164805659623561377642267150191 x^{65} y^{96} v_1^4+397120798757687600238676049258956170932306715 x^{64} y^{97} v_1^4+259344195103331067375307041564357426841427580 x^{63} y^{98} v_1^4+165037215061745267502889661347977800506563480 x^{62} y^{99} v_1^4+102323073334109405303427507797377205532506994 x^{61} y^{100} v_1^4+61799083890657952816088112614980149516838104 x^{60} y^{101} v_1^4+36352402284531481511061648592723893024111900 x^{59} y^{102} v_1^4+20823220722235744148462163930769984143190440 x^{58} y^{103} v_1^4+11612950014654916808690557348907249171193135 x^{57} y^{104} v_1^4+6304172861997635893631671445120079678353481 x^{56} y^{105} v_1^4+3330506415012917163139806207280082312953536 x^{55} y^{106} v_1^4+1711942547588427897326119684242957585821700 x^{54} y^{107} v_1^4+855971272049820971358634328118657298363780 x^{53} y^{108} v_1^4+416206213502785938840548765345467938233130 x^{52} y^{109} v_1^4+196752027195320165065088556297331876232784 x^{51} y^{110} v_1^4+90399579336377195044131713560472538997684 x^{50} y^{111} v_1^4+40356954553944423288040630945394931657945 x^{49} y^{112} v_1^4+17499918004454128872693803494478306802075 x^{48} y^{113} v_1^4+7368386305321939317954391479661457177760 x^{47} y^{114} v_1^4+3011427306201661701481430891700117727176 x^{46} y^{115} v_1^4+1194186605068562122934934212108702563866 x^{45} y^{116} v_1^4+459302490323317254300357171869383554220 x^{44} y^{117} v_1^4+171265306995826885933092196688501391240 x^{43} y^{118} v_1^4+61885767686825334714441754303408529400 x^{42} y^{119} v_1^4+21660010511516469996983202239203785559 x^{41} y^{120} v_1^4+7339338247698536786434418845747519269 x^{40} y^{121} v_1^4+2406338373053751764321925680750878360 x^{39} y^{122} v_1^4+762984377315679132597182527922657960 x^{38} y^{123} v_1^4+233817356855954474044767771096748240 x^{37} y^{124} v_1^4+69209747034055910782239442095170466 x^{36} y^{125} v_1^4+19774133356240567095464516693035992 x^{35} y^{126} v_1^4+5449531975865094660204718202895840 x^{34} y^{127} v_1^4+1447519393181673230828339430100965 x^{33} y^{128} v_1^4+370290993453171979640017028703240 x^{32} y^{129} v_1^4+91146887738190661457782322453712 x^{31} y^{130} v_1^4+21568541888871079711356097919536 x^{30} y^{131} v_1^4+4901755193464025074635360960920 x^{29} y^{132} v_1^4+1068745529071873826346039246400 x^{28} y^{133} v_1^4+223302559062351048635295539760 x^{27} y^{134} v_1^4+44655550330300235607806599536 x^{26} y^{135} v_1^4+8535742612454564782599860172 x^{25} y^{136} v_1^4+1557268815046412363992050825 x^{24} y^{137} v_1^4+270743770022439960886372200 x^{23} y^{138} v_1^4+44779386351502197848922300 x^{22} y^{139} v_1^4+7032360785171863316192160 x^{21} y^{140} v_1^4+1046455480988402703292074 x^{20} y^{141} v_1^4+147207721231333591909080 x^{19} y^{142} v_1^4+19525706592779158557660 x^{18} y^{143} v_1^4+2434924539069087898710 x^{17} y^{144} v_1^4+284534419621696611165 x^{16} y^{145} v_1^4+31039729680508847496 x^{15} y^{146} v_1^4+3147356830386702240 x^{14} y^{147} v_1^4+295131306455284320 x^{13} y^{148} v_1^4+25440351620183500 x^{12} y^{149} v_1^4+2001484590186536 x^{11} y^{150} v_1^4+142465213359040 x^{10} y^{151} v_1^4+9075869851620 x^9 y^{152} v_1^4+510417775725 x^8 y^{153} v_1^4+24891625200 x^7 y^{154} v_1^4+1027567632 x^6 y^{155} v_1^4+34713168 x^5 y^{156} v_1^4+912450 x^4 y^{157} v_1^4+17200 x^3 y^{158} v_1^4+200 x^2 y^{159} v_1^4+x y^{160} v_1^4\right) d^{161}+O[d]^{162}$
$(x+y) d+\left(-x^{42} y v_1-21 x^{41} y^2 v_1-287 x^{40} y^3 v_1-2870 x^{39} y^4 v_1-22386 x^{38} y^5 v_1-141778 x^{37} y^6 v_1-749398 x^{36} y^7 v_1-3372291 x^{35} y^8 v_1-13114465 x^{34} y^9 v_1-44589181 x^{33} y^{10} v_1-133767543 x^{32} y^{11} v_1-356713448 x^{31} y^{12} v_1-850624376 x^{30} y^{13} v_1-1822766520 x^{29} y^{14} v_1-3524015272 x^{28} y^{15} v_1-6167026726 x^{27} y^{16} v_1-9794689506 x^{26} y^{17} v_1-14147884842 x^{25} y^{18} v_1-18615637950 x^{24} y^{19} v_1-22338765540 x^{23} y^{20} v_1-24466267020 x^{22} y^{21} v_1-24466267020 x^{21} y^{22} v_1-22338765540 x^{20} y^{23} v_1-18615637950 x^{19} y^{24} v_1-14147884842 x^{18} y^{25} v_1-9794689506 x^{17} y^{26} v_1-6167026726 x^{16} y^{27} v_1-3524015272 x^{15} y^{28} v_1-1822766520 x^{14} y^{29} v_1-850624376 x^{13} y^{30} v_1-356713448 x^{12} y^{31} v_1-133767543 x^{11} y^{32} v_1-44589181 x^{10} y^{33} v_1-13114465 x^9 y^{34} v_1-3372291 x^8 y^{35} v_1-749398 x^7 y^{36} v_1-141778 x^6 y^{37} v_1-22386 x^5 y^{38} v_1-2870 x^4 y^{39} v_1-287 x^3 y^{40} v_1-21 x^2 y^{41} v_1-x y^{42} v_1\right) d^{43}+\left(x^{84} y v_1^2+63 x^{83} y^2 v_1^2+2030 x^{82} y^3 v_1^2+44485 x^{81} y^4 v_1^2+743043 x^{80} y^5 v_1^2+10049018 x^{79} y^6 v_1^2+114159744 x^{78} y^7 v_1^2+1116429795 x^{77} y^8 v_1^2+9564791600 x^{76} y^9 v_1^2+72737005341 x^{75} y^{10} v_1^2+496067894868 x^{74} y^{11} v_1^2+3059442065134 x^{73} y^{12} v_1^2+17180794528590 x^{72} y^{13} v_1^2+88360194627840 x^{71} y^{14} v_1^2+418241778587048 x^{70} y^{15} v_1^2+1829813948345061 x^{69} y^{16} v_1^2+7426901702678283 x^{68} y^{17} v_1^2+28057198358002800 x^{67} y^{18} v_1^2+98938560193858350 x^{66} y^{19} v_1^2+326497270978498095 x^{65} y^{20} v_1^2+1010586815590189695 x^{64} y^{21} v_1^2+2939888942546818860 x^{63} y^{22} v_1^2+8052739299749617200 x^{62} y^{23} v_1^2+20802909876302149050 x^{61} y^{24} v_1^2+50759100112325128524 x^{60} y^{25} v_1^2+117136384884391139946 x^{59} y^{26} v_1^2+255964692901688406608 x^{58} y^{27} v_1^2+530212578157021428960 x^{57} y^{28} v_1^2+1042141963965623506200 x^{56} y^{29} v_1^2+1945331666070014502616 x^{55} y^{30} v_1^2+3451394891414898572928 x^{54} y^{31} v_1^2+5824228879262775109359 x^{53} y^{32} v_1^2+9354064563664501583000 x^{52} y^{33} v_1^2+14306216391486897888465 x^{51} y^{34} v_1^2+20846201027595197438340 x^{50} y^{35} v_1^2+28953056982771108302648 x^{49} y^{36} v_1^2+38343237625832008434474 x^{48} y^{37} v_1^2+48433563316840431729090 x^{47} y^{38} v_1^2+58368653227987186958440 x^{46} y^{39} v_1^2+67123951212185265002493 x^{45} y^{40} v_1^2+73672629379227729880806 x^{44} y^{41} v_1^2+77180849825857621779893 x^{43} y^{42} v_1^2+77180849825857621779893 x^{42} y^{43} v_1^2+73672629379227729880806 x^{41} y^{44} v_1^2+67123951212185265002493 x^{40} y^{45} v_1^2+58368653227987186958440 x^{39} y^{46} v_1^2+48433563316840431729090 x^{38} y^{47} v_1^2+38343237625832008434474 x^{37} y^{48} v_1^2+28953056982771108302648 x^{36} y^{49} v_1^2+20846201027595197438340 x^{35} y^{50} v_1^2+14306216391486897888465 x^{34} y^{51} v_1^2+9354064563664501583000 x^{33} y^{52} v_1^2+5824228879262775109359 x^{32} y^{53} v_1^2+3451394891414898572928 x^{31} y^{54} v_1^2+1945331666070014502616 x^{30} y^{55} v_1^2+1042141963965623506200 x^{29} y^{56} v_1^2+530212578157021428960 x^{28} y^{57} v_1^2+255964692901688406608 x^{27} y^{58} v_1^2+117136384884391139946 x^{26} y^{59} v_1^2+50759100112325128524 x^{25} y^{60} v_1^2+20802909876302149050 x^{24} y^{61} v_1^2+8052739299749617200 x^{23} y^{62} v_1^2+2939888942546818860 x^{22} y^{63} v_1^2+1010586815590189695 x^{21} y^{64} v_1^2+326497270978498095 x^{20} y^{65} v_1^2+98938560193858350 x^{19} y^{66} v_1^2+28057198358002800 x^{18} y^{67} v_1^2+7426901702678283 x^{17} y^{68} v_1^2+1829813948345061 x^{16} y^{69} v_1^2+418241778587048 x^{15} y^{70} v_1^2+88360194627840 x^{14} y^{71} v_1^2+17180794528590 x^{13} y^{72} v_1^2+3059442065134 x^{12} y^{73} v_1^2+496067894868 x^{11} y^{74} v_1^2+72737005341 x^{10} y^{75} v_1^2+9564791600 x^9 y^{76} v_1^2+1116429795 x^8 y^{77} v_1^2+114159744 x^7 y^{78} v_1^2+10049018 x^6 y^{79} v_1^2+743043 x^5 y^{80} v_1^2+44485 x^4 y^{81} v_1^2+2030 x^3 y^{82} v_1^2+63 x^2 y^{83} v_1^2+x y^{84} v_1^2\right) d^{85}+\left(-x^{126} y v_1^3-126 x^{125} y^2 v_1^3-7280 x^{124} y^3 v_1^3-270165 x^{123} y^4 v_1^3-7389102 x^{122} y^5 v_1^3-160294092 x^{121} y^6 v_1^3-2884957620 x^{120} y^7 v_1^3-44390794095 x^{119} y^8 v_1^3-596509735745 x^{118} y^9 v_1^3-7111551887132 x^{117} y^{10} v_1^3-76137119785272 x^{116} y^{11} v_1^3-739051599989430 x^{115} y^{12} v_1^3-6554944948281240 x^{114} y^{13} v_1^3-53464340487775080 x^{113} y^{14} v_1^3-403182940119825984 x^{112} y^{15} v_1^3-2824110394787126949 x^{111} y^{16} v_1^3-18447206538253918950 x^{110} y^{17} v_1^3-112760986043243063050 x^{109} y^{18} v_1^3-646990911124061956900 x^{108} y^{19} v_1^3-3494077417340913065355 x^{107} y^{20} v_1^3-17804166951362147236980 x^{106} y^{21} v_1^3-85786653381869256233400 x^{105} y^{22} v_1^3-391642774700007223726200 x^{104} y^{23} v_1^3-1697139493276574271629250 x^{103} y^{24} v_1^3-6992265471399598324241034 x^{102} y^{25} v_1^3-27431312447260231663162464 x^{101} y^{26} v_1^3-102613684008147842354310640 x^{100} y^{27} v_1^3-366477973098820451143966960 x^{99} y^{28} v_1^3-1251081019272419988494289960 x^{98} y^{29} v_1^3-4086866608288238032429183152 x^{97} y^{30} v_1^3-12787940903135507193789887952 x^{96} y^{31} v_1^3-38363828533635400844144773215 x^{95} y^{32} v_1^3-110441333920590717609766839225 x^{94} y^{33} v_1^3-305337819851378963702135620440 x^{93} y^{34} v_1^3-811326227879865131146586372652 x^{92} y^{35} v_1^3-2073389277979378984590162366092 x^{91} y^{36} v_1^3-5099416911211439993337542902430 x^{90} y^{37} v_1^3-12077566417092236985271454392740 x^{89} y^{38} v_1^3-27561625984553501732837429034180 x^{88} y^{39} v_1^3-60635577233141655024427608877689 x^{87} y^{40} v_1^3-128665737129363458333500948718829 x^{86} y^{41} v_1^3-263458414199210788318264326299400 x^{85} y^{42} v_1^3-520789888610504501152659197022893 x^{84} y^{43} v_1^3-994235241966453949761577106015418 x^{83} y^{44} v_1^3-1833811668583027902994649704986370 x^{82} y^{45} v_1^3-3268968626662896654218449704539220 x^{81} y^{46} v_1^3-5633754441744063967395445241602995 x^{80} y^{47} v_1^3-9389590736278449849951574076534394 x^{79} y^{48} v_1^3-15138319758518698733435308888962744 x^{78} y^{49} v_1^3-23615778823310016225186676886131020 x^{77} y^{50} v_1^3-35655195478345115026967448354104040 x^{76} y^{51} v_1^3-52111439545282983719362228885962720 x^{75} y^{52} v_1^3-73742603130123254020429483740615084 x^{74} y^{53} v_1^3-101054678363505688385849657264893224 x^{73} y^{54} v_1^3-134127118555200404461971554397528990 x^{72} y^{55} v_1^3-172449152428115847878762445970221000 x^{71} y^{56} v_1^3-214805084603443077570224825268220440 x^{70} y^{57} v_1^3-259247515900707418548942226589688288 x^{69} y^{58} v_1^3-303187772833030826962668892395422703 x^{68} y^{59} v_1^3-343612809210768321308374368443572200 x^{67} y^{60} v_1^3-377410790444614406471454964756790850 x^{66} y^{61} v_1^3-401759873699105666449289020735964700 x^{65} y^{62} v_1^3-414514155403839182272928049439443135 x^{64} y^{63} v_1^3-414514155403839182272928049439443135 x^{63} y^{64} v_1^3-401759873699105666449289020735964700 x^{62} y^{65} v_1^3-377410790444614406471454964756790850 x^{61} y^{66} v_1^3-343612809210768321308374368443572200 x^{60} y^{67} v_1^3-303187772833030826962668892395422703 x^{59} y^{68} v_1^3-259247515900707418548942226589688288 x^{58} y^{69} v_1^3-214805084603443077570224825268220440 x^{57} y^{70} v_1^3-172449152428115847878762445970221000 x^{56} y^{71} v_1^3-134127118555200404461971554397528990 x^{55} y^{72} v_1^3-101054678363505688385849657264893224 x^{54} y^{73} v_1^3-73742603130123254020429483740615084 x^{53} y^{74} v_1^3-52111439545282983719362228885962720 x^{52} y^{75} v_1^3-35655195478345115026967448354104040 x^{51} y^{76} v_1^3-23615778823310016225186676886131020 x^{50} y^{77} v_1^3-15138319758518698733435308888962744 x^{49} y^{78} v_1^3-9389590736278449849951574076534394 x^{48} y^{79} v_1^3-5633754441744063967395445241602995 x^{47} y^{80} v_1^3-3268968626662896654218449704539220 x^{46} y^{81} v_1^3-1833811668583027902994649704986370 x^{45} y^{82} v_1^3-994235241966453949761577106015418 x^{44} y^{83} v_1^3-520789888610504501152659197022893 x^{43} y^{84} v_1^3-263458414199210788318264326299400 x^{42} y^{85} v_1^3-128665737129363458333500948718829 x^{41} y^{86} v_1^3-60635577233141655024427608877689 x^{40} y^{87} v_1^3-27561625984553501732837429034180 x^{39} y^{88} v_1^3-12077566417092236985271454392740 x^{38} y^{89} v_1^3-5099416911211439993337542902430 x^{37} y^{90} v_1^3-2073389277979378984590162366092 x^{36} y^{91} v_1^3-811326227879865131146586372652 x^{35} y^{92} v_1^3-305337819851378963702135620440 x^{34} y^{93} v_1^3-110441333920590717609766839225 x^{33} y^{94} v_1^3-38363828533635400844144773215 x^{32} y^{95} v_1^3-12787940903135507193789887952 x^{31} y^{96} v_1^3-4086866608288238032429183152 x^{30} y^{97} v_1^3-1251081019272419988494289960 x^{29} y^{98} v_1^3-366477973098820451143966960 x^{28} y^{99} v_1^3-102613684008147842354310640 x^{27} y^{100} v_1^3-27431312447260231663162464 x^{26} y^{101} v_1^3-6992265471399598324241034 x^{25} y^{102} v_1^3-1697139493276574271629250 x^{24} y^{103} v_1^3-391642774700007223726200 x^{23} y^{104} v_1^3-85786653381869256233400 x^{22} y^{105} v_1^3-17804166951362147236980 x^{21} y^{106} v_1^3-3494077417340913065355 x^{20} y^{107} v_1^3-646990911124061956900 x^{19} y^{108} v_1^3-112760986043243063050 x^{18} y^{109} v_1^3-18447206538253918950 x^{17} y^{110} v_1^3-2824110394787126949 x^{16} y^{111} v_1^3-403182940119825984 x^{15} y^{112} v_1^3-53464340487775080 x^{14} y^{113} v_1^3-6554944948281240 x^{13} y^{114} v_1^3-739051599989430 x^{12} y^{115} v_1^3-76137119785272 x^{11} y^{116} v_1^3-7111551887132 x^{10} y^{117} v_1^3-596509735745 x^9 y^{118} v_1^3-44390794095 x^8 y^{119} v_1^3-2884957620 x^7 y^{120} v_1^3-160294092 x^6 y^{121} v_1^3-7389102 x^5 y^{122} v_1^3-270165 x^4 y^{123} v_1^3-7280 x^3 y^{124} v_1^3-126 x^2 y^{125} v_1^3-x y^{126} v_1^3\right) d^{127}+O[d]^{128}$
$(x+y) d+\left(-x^{46} y v_1-23 x^{45} y^2 v_1-345 x^{44} y^3 v_1-3795 x^{43} y^4 v_1-32637 x^{42} y^5 v_1-228459 x^{41} y^6 v_1-1338117 x^{40} y^7 v_1-6690585 x^{39} y^8 v_1-28992535 x^{38} y^9 v_1-110171633 x^{37} y^{10} v_1-370577311 x^{36} y^{11} v_1-1111731933 x^{35} y^{12} v_1-2993124435 x^{34} y^{13} v_1-7269016485 x^{33} y^{14} v_1-15991836267 x^{32} y^{15} v_1-31983672534 x^{31} y^{16} v_1-58323167562 x^{30} y^{17} v_1-97205279270 x^{29} y^{18} v_1-148365952570 x^{28} y^{19} v_1-207712333598 x^{27} y^{20} v_1-267058714626 x^{26} y^{21} v_1-315614844558 x^{25} y^{22} v_1-343059613650 x^{24} y^{23} v_1-343059613650 x^{23} y^{24} v_1-315614844558 x^{22} y^{25} v_1-267058714626 x^{21} y^{26} v_1-207712333598 x^{20} y^{27} v_1-148365952570 x^{19} y^{28} v_1-97205279270 x^{18} y^{29} v_1-58323167562 x^{17} y^{30} v_1-31983672534 x^{16} y^{31} v_1-15991836267 x^{15} y^{32} v_1-7269016485 x^{14} y^{33} v_1-2993124435 x^{13} y^{34} v_1-1111731933 x^{12} y^{35} v_1-370577311 x^{11} y^{36} v_1-110171633 x^{10} y^{37} v_1-28992535 x^9 y^{38} v_1-6690585 x^8 y^{39} v_1-1338117 x^7 y^{40} v_1-228459 x^6 y^{41} v_1-32637 x^5 y^{42} v_1-3795 x^4 y^{43} v_1-345 x^3 y^{44} v_1-23 x^2 y^{45} v_1-x y^{46} v_1\right) d^{47}+\left(x^{92} y v_1^2+69 x^{91} y^2 v_1^2+2438 x^{90} y^3 v_1^2+58650 x^{89} y^4 v_1^2+1076607 x^{88} y^5 v_1^2+16018695 x^{87} y^6 v_1^2+200427612 x^{86} y^7 v_1^2+2161287414 x^{85} y^8 v_1^2+20441151445 x^{84} y^9 v_1^2+171815843771 x^{83} y^{10} v_1^2+1296799216674 x^{82} y^{11} v_1^2+8862573045872 x^{81} y^{12} v_1^2+55223640564099 x^{80} y^{13} v_1^2+315570929382765 x^{79} y^{14} v_1^2+1662022886585496 x^{78} y^{15} v_1^2+8102393555776827 x^{77} y^{16} v_1^2+36699135016980249 x^{76} y^{17} v_1^2+154952000610306988 x^{75} y^{18} v_1^2+611652782354006470 x^{74} y^{19} v_1^2+2263115502422157537 x^{73} y^{20} v_1^2+7867020823097643207 x^{72} y^{21} v_1^2+25746613918479858690 x^{71} y^{22} v_1^2+79478678091410481780 x^{70} y^{23} v_1^2+231812811443006852175 x^{69} y^{24} v_1^2+639803359898313756561 x^{68} y^{25} v_1^2+1673331864616494693324 x^{67} y^{26} v_1^2+4152342034626421387402 x^{66} y^{27} v_1^2+9787663367482073508589 x^{65} y^{28} v_1^2+21937866168591507970935 x^{64} y^{29} v_1^2+46800781159720206838890 x^{63} y^{30} v_1^2+95111264937527887893504 x^{62} y^{31} v_1^2+184278075816476274629931 x^{61} y^{32} v_1^2+340635231054705837271812 x^{60} y^{33} v_1^2+601120995978895647133515 x^{59} y^{34} v_1^2+1013318250364425202614144 x^{58} y^{35} v_1^2+1632568292253796530344543 x^{57} y^{36} v_1^2+2515037639418010981242956 x^{56} y^{37} v_1^2+3706371258089700422403207 x^{55} y^{38} v_1^2+5226933825511115987002800 x^{54} y^{39} v_1^2+7056360664440006583791897 x^{53} y^{40} v_1^2+9121636956471228023178960 x^{52} y^{41} v_1^2+11293455279440568028730397 x^{51} y^{42} v_1^2+13394563238406255103846824 x^{50} y^{43} v_1^2+15221094589098017163462645 x^{49} y^{44} v_1^2+16574080774795618689103792 x^{48} y^{45} v_1^2+17294692982395428197325697 x^{47} y^{46} v_1^2+17294692982395428197325697 x^{46} y^{47} v_1^2+16574080774795618689103792 x^{45} y^{48} v_1^2+15221094589098017163462645 x^{44} y^{49} v_1^2+13394563238406255103846824 x^{43} y^{50} v_1^2+11293455279440568028730397 x^{42} y^{51} v_1^2+9121636956471228023178960 x^{41} y^{52} v_1^2+7056360664440006583791897 x^{40} y^{53} v_1^2+5226933825511115987002800 x^{39} y^{54} v_1^2+3706371258089700422403207 x^{38} y^{55} v_1^2+2515037639418010981242956 x^{37} y^{56} v_1^2+1632568292253796530344543 x^{36} y^{57} v_1^2+1013318250364425202614144 x^{35} y^{58} v_1^2+601120995978895647133515 x^{34} y^{59} v_1^2+340635231054705837271812 x^{33} y^{60} v_1^2+184278075816476274629931 x^{32} y^{61} v_1^2+95111264937527887893504 x^{31} y^{62} v_1^2+46800781159720206838890 x^{30} y^{63} v_1^2+21937866168591507970935 x^{29} y^{64} v_1^2+9787663367482073508589 x^{28} y^{65} v_1^2+4152342034626421387402 x^{27} y^{66} v_1^2+1673331864616494693324 x^{26} y^{67} v_1^2+639803359898313756561 x^{25} y^{68} v_1^2+231812811443006852175 x^{24} y^{69} v_1^2+79478678091410481780 x^{23} y^{70} v_1^2+25746613918479858690 x^{22} y^{71} v_1^2+7867020823097643207 x^{21} y^{72} v_1^2+2263115502422157537 x^{20} y^{73} v_1^2+611652782354006470 x^{19} y^{74} v_1^2+154952000610306988 x^{18} y^{75} v_1^2+36699135016980249 x^{17} y^{76} v_1^2+8102393555776827 x^{16} y^{77} v_1^2+1662022886585496 x^{15} y^{78} v_1^2+315570929382765 x^{14} y^{79} v_1^2+55223640564099 x^{13} y^{80} v_1^2+8862573045872 x^{12} y^{81} v_1^2+1296799216674 x^{11} y^{82} v_1^2+171815843771 x^{10} y^{83} v_1^2+20441151445 x^9 y^{84} v_1^2+2161287414 x^8 y^{85} v_1^2+200427612 x^7 y^{86} v_1^2+16018695 x^6 y^{87} v_1^2+1076607 x^5 y^{88} v_1^2+58650 x^4 y^{89} v_1^2+2438 x^3 y^{90} v_1^2+69 x^2 y^{91} v_1^2+x y^{92} v_1^2\right) d^{93}+\left(-x^{138} y v_1^3-138 x^{137} y^2 v_1^3-8740 x^{136} y^3 v_1^3-355810 x^{135} y^4 v_1^3-10683477 x^{134} y^5 v_1^3-254616348 x^{133} y^6 v_1^3-5038138224 x^{132} y^7 v_1^3-85290568110 x^{131} y^8 v_1^3-1261892753935 x^{130} y^9 v_1^3-16576421644926 x^{129} y^{10} v_1^3-195693016688988 x^{128} y^{11} v_1^3-2096254751061744 x^{127} y^{12} v_1^3-20534020054782675 x^{126} y^{13} v_1^3-185121751422426840 x^{125} y^{14} v_1^3-1544343284740142496 x^{124} y^{15} v_1^3-11976762850291881171 x^{123} y^{16} v_1^3-86692100934187649898 x^{122} y^{17} v_1^3-587734747221215489630 x^{121} y^{18} v_1^3-3743553990349042124640 x^{120} y^{19} v_1^3-22463587057596674905377 x^{119} y^{20} v_1^3-127301527013870922107010 x^{118} y^{21} v_1^3-682824846051953425705380 x^{117} y^{22} v_1^3-3473579782507593619504800 x^{116} y^{23} v_1^3-16789200761598145501125375 x^{115} y^{24} v_1^3-77230963306711367618933286 x^{114} y^{25} v_1^3-338629743215137536054631578 x^{113} y^{26} v_1^3-1417232336909091721761141784 x^{112} y^{27} v_1^3-5668939135299734369118075725 x^{111} y^{28} v_1^3-21698375179875496693994398710 x^{110} y^{29} v_1^3-79560755793657980931519634160 x^{109} y^{30} v_1^3-279745978385739773706456929744 x^{108} y^{31} v_1^3-944142861329947552735566767817 x^{107} y^{32} v_1^3-3061312042523242816606008306855 x^{106} y^{33} v_1^3-9544091086634635348314378913710 x^{105} y^{34} v_1^3-28632274273222156409368339355274 x^{104} y^{35} v_1^3-82715460644098966325305066259779 x^{103} y^{36} v_1^3-230261960524286113242508868398557 x^{102} y^{37} v_1^3-618071581955770825214329490209860 x^{101} y^{38} v_1^3-1600646922599571090809764153956540 x^{100} y^{39} v_1^3-4001617313555288391464416968683247 x^{99} y^{40} v_1^3-9662441815023430877324332410975093 x^{98} y^{41} v_1^3-22545697579681460659864010321005614 x^{97} y^{42} v_1^3-50858899204769021006006464432626930 x^{96} y^{43} v_1^3-110964871007444413147657575925557765 x^{95} y^{44} v_1^3-234259172143401175197628278976392407 x^{94} y^{45} v_1^3-478703525701636224907983650453431920 x^{93} y^{46} v_1^3-947221870022659989076916268456244177 x^{92} y^{47} v_1^3-1815508584226672393172218466563571796 x^{91} y^{48} v_1^3-3371658799293326967623218026495810138 x^{90} y^{49} v_1^3-6068985838741383104960198702796300684 x^{89} y^{50} v_1^3-10590975287226648285504101049379035123 x^{88} y^{51} v_1^3-17923188947623449504732642234662801064 x^{87} y^{52} v_1^3-29421083744219133849565192354022849748 x^{86} y^{53} v_1^3-46855800037094958620170076716519008424 x^{85} y^{54} v_1^3-72413509148241369693339117349884335677 x^{84} y^{55} v_1^3-108620263722364569577648094005146019304 x^{83} y^{56} v_1^3-158166348929058812830481583413484810670 x^{82} y^{57} v_1^3-223614493313497955595827773624654660828 x^{81} y^{58} v_1^3-306996168786328302871200198314776130319 x^{80} y^{59} v_1^3-409328225048438077796831245494018026772 x^{79} y^{60} v_1^3-530113602931583924375610955229127136104 x^{78} y^{61} v_1^3-666917113365541161261225016077674392656 x^{77} y^{62} v_1^3-815120916335661466120046276445240702681 x^{76} y^{63} v_1^3-967956088148598012955377541647398641323 x^{75} y^{64} v_1^3-1116872409402228486274458663228367740514 x^{74} y^{65} v_1^3-1252250883269165276641200803914376180118 x^{73} y^{66} v_1^3-1364392753412672616518741584826534150763 x^{72} y^{67} v_1^3-1444651150672241594592494074725481823385 x^{71} y^{68} v_1^3-1486525097068538452612089184157458149180 x^{70} y^{69} v_1^3-1486525097068538452612089184157458149180 x^{69} y^{70} v_1^3-1444651150672241594592494074725481823385 x^{68} y^{71} v_1^3-1364392753412672616518741584826534150763 x^{67} y^{72} v_1^3-1252250883269165276641200803914376180118 x^{66} y^{73} v_1^3-1116872409402228486274458663228367740514 x^{65} y^{74} v_1^3-967956088148598012955377541647398641323 x^{64} y^{75} v_1^3-815120916335661466120046276445240702681 x^{63} y^{76} v_1^3-666917113365541161261225016077674392656 x^{62} y^{77} v_1^3-530113602931583924375610955229127136104 x^{61} y^{78} v_1^3-409328225048438077796831245494018026772 x^{60} y^{79} v_1^3-306996168786328302871200198314776130319 x^{59} y^{80} v_1^3-223614493313497955595827773624654660828 x^{58} y^{81} v_1^3-158166348929058812830481583413484810670 x^{57} y^{82} v_1^3-108620263722364569577648094005146019304 x^{56} y^{83} v_1^3-72413509148241369693339117349884335677 x^{55} y^{84} v_1^3-46855800037094958620170076716519008424 x^{54} y^{85} v_1^3-29421083744219133849565192354022849748 x^{53} y^{86} v_1^3-17923188947623449504732642234662801064 x^{52} y^{87} v_1^3-10590975287226648285504101049379035123 x^{51} y^{88} v_1^3-6068985838741383104960198702796300684 x^{50} y^{89} v_1^3-3371658799293326967623218026495810138 x^{49} y^{90} v_1^3-1815508584226672393172218466563571796 x^{48} y^{91} v_1^3-947221870022659989076916268456244177 x^{47} y^{92} v_1^3-478703525701636224907983650453431920 x^{46} y^{93} v_1^3-234259172143401175197628278976392407 x^{45} y^{94} v_1^3-110964871007444413147657575925557765 x^{44} y^{95} v_1^3-50858899204769021006006464432626930 x^{43} y^{96} v_1^3-22545697579681460659864010321005614 x^{42} y^{97} v_1^3-9662441815023430877324332410975093 x^{41} y^{98} v_1^3-4001617313555288391464416968683247 x^{40} y^{99} v_1^3-1600646922599571090809764153956540 x^{39} y^{100} v_1^3-618071581955770825214329490209860 x^{38} y^{101} v_1^3-230261960524286113242508868398557 x^{37} y^{102} v_1^3-82715460644098966325305066259779 x^{36} y^{103} v_1^3-28632274273222156409368339355274 x^{35} y^{104} v_1^3-9544091086634635348314378913710 x^{34} y^{105} v_1^3-3061312042523242816606008306855 x^{33} y^{106} v_1^3-944142861329947552735566767817 x^{32} y^{107} v_1^3-279745978385739773706456929744 x^{31} y^{108} v_1^3-79560755793657980931519634160 x^{30} y^{109} v_1^3-21698375179875496693994398710 x^{29} y^{110} v_1^3-5668939135299734369118075725 x^{28} y^{111} v_1^3-1417232336909091721761141784 x^{27} y^{112} v_1^3-338629743215137536054631578 x^{26} y^{113} v_1^3-77230963306711367618933286 x^{25} y^{114} v_1^3-16789200761598145501125375 x^{24} y^{115} v_1^3-3473579782507593619504800 x^{23} y^{116} v_1^3-682824846051953425705380 x^{22} y^{117} v_1^3-127301527013870922107010 x^{21} y^{118} v_1^3-22463587057596674905377 x^{20} y^{119} v_1^3-3743553990349042124640 x^{19} y^{120} v_1^3-587734747221215489630 x^{18} y^{121} v_1^3-86692100934187649898 x^{17} y^{122} v_1^3-11976762850291881171 x^{16} y^{123} v_1^3-1544343284740142496 x^{15} y^{124} v_1^3-185121751422426840 x^{14} y^{125} v_1^3-20534020054782675 x^{13} y^{126} v_1^3-2096254751061744 x^{12} y^{127} v_1^3-195693016688988 x^{11} y^{128} v_1^3-16576421644926 x^{10} y^{129} v_1^3-1261892753935 x^9 y^{130} v_1^3-85290568110 x^8 y^{131} v_1^3-5038138224 x^7 y^{132} v_1^3-254616348 x^6 y^{133} v_1^3-10683477 x^5 y^{134} v_1^3-355810 x^4 y^{135} v_1^3-8740 x^3 y^{136} v_1^3-138 x^2 y^{137} v_1^3-x y^{138} v_1^3\right) d^{139}+O[d]^{140}$
$(x+y) d+x y v_1 d^2+\left(x^2 y v_1^2+x y^2 v_1^2\right) d^3+\left(x^3 y \left(\frac{6 v_1^3}{7}+\frac{2 v_2}{7}\right)+x y^3 \left(\frac{6 v_1^3}{7}+\frac{2 v_2}{7}\right)+x^2 y^2 \left(\frac{16 v_1^3}{7}+\frac{3 v_2}{7}\right)\right) d^4+\left(x^4 y \left(\frac{5 v_1^4}{7}+\frac{4 v_1 v_2}{7}\right)+x y^4 \left(\frac{5 v_1^4}{7}+\frac{4 v_1 v_2}{7}\right)+x^3 y^2 \left(\frac{26 v_1^4}{7}+\frac{11 v_1 v_2}{7}\right)+x^2 y^3 \left(\frac{26 v_1^4}{7}+\frac{11 v_1 v_2}{7}\right)\right) d^5+\left(x^5 y \left(\frac{4 v_1^5}{7}+\frac{6}{7} v_1^2 v_2\right)+x y^5 \left(\frac{4 v_1^5}{7}+\frac{6}{7} v_1^2 v_2\right)+x^4 y^2 \left(5 v_1^5+4 v_1^2 v_2\right)+x^2 y^4 \left(5 v_1^5+4 v_1^2 v_2\right)+x^3 y^3 \left(\frac{66 v_1^5}{7}+\frac{43}{7} v_1^2 v_2\right)\right) d^6+\left(x^6 y \left(\frac{22 v_1^6}{49}+\frac{52}{49} v_1^3 v_2+\frac{4 v_2^2}{49}\right)+x y^6 \left(\frac{22 v_1^6}{49}+\frac{52}{49} v_1^3 v_2+\frac{4 v_2^2}{49}\right)+x^5 y^2 \left(\frac{295 v_1^6}{49}+\frac{381}{49} v_1^3 v_2+\frac{18 v_2^2}{49}\right)+x^2 y^5 \left(\frac{295 v_1^6}{49}+\frac{381}{49} v_1^3 v_2+\frac{18 v_2^2}{49}\right)+x^4 y^3 \left(\frac{901 v_1^6}{49}+\frac{876}{49} v_1^3 v_2+\frac{34 v_2^2}{49}\right)+x^3 y^4 \left(\frac{901 v_1^6}{49}+\frac{876}{49} v_1^3 v_2+\frac{34 v_2^2}{49}\right)\right) d^7+\left(x^7 y \left(\frac{2166 v_1^7}{6223}+\frac{7352 v_1^4 v_2}{6223}+\frac{1426 v_1 v_2^2}{6223}+\frac{4 v_3}{127}\right)+x y^7 \left(\frac{2166 v_1^7}{6223}+\frac{7352 v_1^4 v_2}{6223}+\frac{1426 v_1 v_2^2}{6223}+\frac{4 v_3}{127}\right)+x^6 y^2 \left(\frac{41744 v_1^7}{6223}+\frac{79326 v_1^4 v_2}{6223}+\frac{10071 v_1 v_2^2}{6223}+\frac{14 v_3}{127}\right)+x^2 y^6 \left(\frac{41744 v_1^7}{6223}+\frac{79326 v_1^4 v_2}{6223}+\frac{10071 v_1 v_2^2}{6223}+\frac{14 v_3}{127}\right)+x^5 y^3 \left(\frac{189025 v_1^7}{6223}+\frac{261903 v_1^4 v_2}{6223}+\frac{26238 v_1 v_2^2}{6223}+\frac{28 v_3}{127}\right)+x^3 y^5 \left(\frac{189025 v_1^7}{6223}+\frac{261903 v_1^4 v_2}{6223}+\frac{26238 v_1 v_2^2}{6223}+\frac{28 v_3}{127}\right)+x^4 y^4 \left(\frac{303242 v_1^7}{6223}+\frac{378909 v_1^4 v_2}{6223}+\frac{35274 v_1 v_2^2}{6223}+\frac{35 v_3}{127}\right)\right) d^8+\left(x^8 y \left(\frac{1665 v_1^8}{6223}+\frac{7592 v_1^5 v_2}{6223}+\frac{2852 v_1^2 v_2^2}{6223}+\frac{8 v_1 v_3}{127}\right)+x y^8 \left(\frac{1665 v_1^8}{6223}+\frac{7592 v_1^5 v_2}{6223}+\frac{2852 v_1^2 v_2^2}{6223}+\frac{8 v_1 v_3}{127}\right)+x^7 y^2 \left(\frac{43959 v_1^8}{6223}+\frac{115028 v_1^5 v_2}{6223}+\frac{28591 v_1^2 v_2^2}{6223}+\frac{46 v_1 v_3}{127}\right)+x^2 y^7 \left(\frac{43959 v_1^8}{6223}+\frac{115028 v_1^5 v_2}{6223}+\frac{28591 v_1^2 v_2^2}{6223}+\frac{46 v_1 v_3}{127}\right)+x^6 y^3 \left(\frac{278160 v_1^8}{6223}+\frac{524450 v_1^5 v_2}{6223}+\frac{100545 v_1^2 v_2^2}{6223}+\frac{126 v_1 v_3}{127}\right)+x^3 y^6 \left(\frac{278160 v_1^8}{6223}+\frac{524450 v_1^5 v_2}{6223}+\frac{100545 v_1^2 v_2^2}{6223}+\frac{126 v_1 v_3}{127}\right)+x^5 y^4 \left(\frac{653421 v_1^8}{6223}+\frac{1051929 v_1^5 v_2}{6223}+\frac{178097 v_1^2 v_2^2}{6223}+\frac{203 v_1 v_3}{127}\right)+x^4 y^5 \left(\frac{653421 v_1^8}{6223}+\frac{1051929 v_1^5 v_2}{6223}+\frac{178097 v_1^2 v_2^2}{6223}+\frac{203 v_1 v_3}{127}\right)\right) d^9+\left(x^9 y \left(\frac{8910 v_1^9}{43561}+\frac{52030 v_1^6 v_2}{43561}+\frac{31978 v_1^3 v_2^2}{43561}+\frac{8 v_2^3}{343}+\frac{12}{127} v_1^2 v_3\right)+x y^9 \left(\frac{8910 v_1^9}{43561}+\frac{52030 v_1^6 v_2}{43561}+\frac{31978 v_1^3 v_2^2}{43561}+\frac{8 v_2^3}{343}+\frac{12}{127} v_1^2 v_3\right)+x^8 y^2 \left(\frac{310392 v_1^9}{43561}+\frac{1065776 v_1^6 v_2}{43561}+\frac{437840 v_1^3 v_2^2}{43561}+\frac{72 v_2^3}{343}+\frac{104}{127} v_1^2 v_3\right)+x^2 y^8 \left(\frac{310392 v_1^9}{43561}+\frac{1065776 v_1^6 v_2}{43561}+\frac{437840 v_1^3 v_2^2}{43561}+\frac{72 v_2^3}{343}+\frac{104}{127} v_1^2 v_3\right)+x^7 y^3 \left(\frac{2629325 v_1^9}{43561}+\frac{6489104 v_1^6 v_2}{43561}+\frac{2036337 v_1^3 v_2^2}{43561}+\frac{260 v_2^3}{343}+\frac{382}{127} v_1^2 v_3\right)+x^3 y^7 \left(\frac{2629325 v_1^9}{43561}+\frac{6489104 v_1^6 v_2}{43561}+\frac{2036337 v_1^3 v_2^2}{43561}+\frac{260 v_2^3}{343}+\frac{382}{127} v_1^2 v_3\right)+x^6 y^4 \left(\frac{8522341 v_1^9}{43561}+\frac{17427944 v_1^6 v_2}{43561}+\frac{4678215 v_1^3 v_2^2}{43561}+\frac{523 v_2^3}{343}+\frac{791}{127} v_1^2 v_3\right)+x^4 y^6 \left(\frac{8522341 v_1^9}{43561}+\frac{17427944 v_1^6 v_2}{43561}+\frac{4678215 v_1^3 v_2^2}{43561}+\frac{523 v_2^3}{343}+\frac{791}{127} v_1^2 v_3\right)+x^5 y^5 \left(\frac{12430100 v_1^9}{43561}+\frac{23899418 v_1^6 v_2}{43561}+\frac{6095207 v_1^3 v_2^2}{43561}+\frac{654 v_2^3}{343}+\frac{1001}{127} v_1^2 v_3\right)\right) d^{10}+\left(x^{10} y \left(\frac{6786 v_1^{10}}{43561}+\frac{48940 v_1^7 v_2}{43561}+\frac{44640 v_1^4 v_2^2}{43561}+\frac{3672 v_1 v_2^3}{43561}+\frac{104}{889} v_1^3 v_3+\frac{16 v_2 v_3}{889}\right)+x y^{10} \left(\frac{6786 v_1^{10}}{43561}+\frac{48940 v_1^7 v_2}{43561}+\frac{44640 v_1^4 v_2^2}{43561}+\frac{3672 v_1 v_2^3}{43561}+\frac{104}{889} v_1^3 v_3+\frac{16 v_2 v_3}{889}\right)+x^9 y^2 \left(\frac{302690 v_1^{10}}{43561}+\frac{1314435 v_1^7 v_2}{43561}+\frac{808011 v_1^4 v_2^2}{43561}+\frac{47352 v_1 v_2^3}{43561}+\frac{1298}{889} v_1^3 v_3+\frac{120 v_2 v_3}{889}\right)+x^2 y^9 \left(\frac{302690 v_1^{10}}{43561}+\frac{1314435 v_1^7 v_2}{43561}+\frac{808011 v_1^4 v_2^2}{43561}+\frac{47352 v_1 v_2^3}{43561}+\frac{1298}{889} v_1^3 v_3+\frac{120 v_2 v_3}{889}\right)+x^8 y^3 \left(\frac{474526 v_1^{10}}{6223}+\frac{1484628 v_1^7 v_2}{6223}+\frac{99212}{889} v_1^4 v_2^2+\frac{32044 v_1 v_2^3}{6223}+\frac{6312}{889} v_1^3 v_3+\frac{424 v_2 v_3}{889}\right)+x^3 y^8 \left(\frac{474526 v_1^{10}}{6223}+\frac{1484628 v_1^7 v_2}{6223}+\frac{99212}{889} v_1^4 v_2^2+\frac{32044 v_1 v_2^3}{6223}+\frac{6312}{889} v_1^3 v_3+\frac{424 v_2 v_3}{889}\right)+x^7 y^4 \left(\frac{14241679 v_1^{10}}{43561}+\frac{36308759 v_1^7 v_2}{43561}+\frac{14247855 v_1^4 v_2^2}{43561}+\frac{567297 v_1 v_2^3}{43561}+\frac{16670}{889} v_1^3 v_3+\frac{918 v_2 v_3}{889}\right)+x^4 y^7 \left(\frac{14241679 v_1^{10}}{43561}+\frac{36308759 v_1^7 v_2}{43561}+\frac{14247855 v_1^4 v_2^2}{43561}+\frac{567297 v_1 v_2^3}{43561}+\frac{16670}{889} v_1^3 v_3+\frac{918 v_2 v_3}{889}\right)+x^6 y^5 \left(\frac{4071629 v_1^{10}}{6223}+\frac{1342190}{889} v_1^7 v_2+\frac{3381473 v_1^4 v_2^2}{6223}+\frac{125647 v_1 v_2^3}{6223}+\frac{3797}{127} v_1^3 v_3+\frac{190 v_2 v_3}{127}\right)+x^5 y^6 \left(\frac{4071629 v_1^{10}}{6223}+\frac{1342190}{889} v_1^7 v_2+\frac{3381473 v_1^4 v_2^2}{6223}+\frac{125647 v_1 v_2^3}{6223}+\frac{3797}{127} v_1^3 v_3+\frac{190 v_2 v_3}{127}\right)\right) d^{11}+\left(x^{11} y \left(\frac{5156 v_1^{11}}{43561}+\frac{44636 v_1^8 v_2}{43561}+\frac{56426 v_1^5 v_2^2}{43561}+\frac{8984 v_1^2 v_2^3}{43561}+\frac{116}{889} v_1^4 v_3+\frac{48}{889} v_1 v_2 v_3\right)+x y^{11} \left(\frac{5156 v_1^{11}}{43561}+\frac{44636 v_1^8 v_2}{43561}+\frac{56426 v_1^5 v_2^2}{43561}+\frac{8984 v_1^2 v_2^3}{43561}+\frac{116}{889} v_1^4 v_3+\frac{48}{889} v_1 v_2 v_3\right)+x^{10} y^2 \left(\frac{287271 v_1^{11}}{43561}+\frac{1532536 v_1^8 v_2}{43561}+\frac{10371}{343} v_1^5 v_2^2+\frac{152868 v_1^2 v_2^3}{43561}+\frac{1994}{889} v_1^4 v_3+\frac{520}{889} v_1 v_2 v_3\right)+x^2 y^{10} \left(\frac{287271 v_1^{11}}{43561}+\frac{1532536 v_1^8 v_2}{43561}+\frac{10371}{343} v_1^5 v_2^2+\frac{152868 v_1^2 v_2^3}{43561}+\frac{1994}{889} v_1^4 v_3+\frac{520}{889} v_1 v_2 v_3\right)+x^9 y^3 \left(\frac{3978654 v_1^{11}}{43561}+\frac{15369299 v_1^8 v_2}{43561}+\frac{10048631 v_1^5 v_2^2}{43561}+\frac{921640 v_1^2 v_2^3}{43561}+\frac{12590}{889} v_1^4 v_3+\frac{2400}{889} v_1 v_2 v_3\right)+x^3 y^9 \left(\frac{3978654 v_1^{11}}{43561}+\frac{15369299 v_1^8 v_2}{43561}+\frac{10048631 v_1^5 v_2^2}{43561}+\frac{921640 v_1^2 v_2^3}{43561}+\frac{12590}{889} v_1^4 v_3+\frac{2400}{889} v_1 v_2 v_3\right)+x^8 y^4 \left(\frac{21859212 v_1^{11}}{43561}+\frac{68187074 v_1^8 v_2}{43561}+\frac{36964465 v_1^5 v_2^2}{43561}+\frac{2890247 v_1^2 v_2^3}{43561}+\frac{41894}{889} v_1^4 v_3+\frac{6470}{889} v_1 v_2 v_3\right)+x^4 y^8 \left(\frac{21859212 v_1^{11}}{43561}+\frac{68187074 v_1^8 v_2}{43561}+\frac{36964465 v_1^5 v_2^2}{43561}+\frac{2890247 v_1^2 v_2^3}{43561}+\frac{41894}{889} v_1^4 v_3+\frac{6470}{889} v_1 v_2 v_3\right)+x^7 y^5 \left(\frac{57510694 v_1^{11}}{43561}+\frac{158500527 v_1^8 v_2}{43561}+\frac{76941264 v_1^5 v_2^2}{43561}+\frac{5492099 v_1^2 v_2^3}{43561}+\frac{83320}{889} v_1^4 v_3+\frac{11384}{889} v_1 v_2 v_3\right)+x^5 y^7 \left(\frac{57510694 v_1^{11}}{43561}+\frac{158500527 v_1^8 v_2}{43561}+\frac{76941264 v_1^5 v_2^2}{43561}+\frac{5492099 v_1^2 v_2^3}{43561}+\frac{83320}{889} v_1^4 v_3+\frac{11384}{889} v_1 v_2 v_3\right)+x^6 y^6 \left(\frac{78755046 v_1^{11}}{43561}+\frac{208387871 v_1^8 v_2}{43561}+\frac{97525934 v_1^5 v_2^2}{43561}+\frac{6759509 v_1^2 v_2^3}{43561}+\frac{14895}{127} v_1^4 v_3+\frac{1955}{127} v_1 v_2 v_3\right)\right) d^{12}+O[d]^{13}$
$(x+y) d+\frac{1}{8} \left(x^2 y v_1+x y^2 v_1\right) d^3+\frac{1}{64} \left(x^4 y v_1^2+3 x^3 y^2 v_1^2+3 x^2 y^3 v_1^2+x y^4 v_1^2\right) d^5+\frac{1}{512} \left(x^6 y v_1^3+6 x^5 y^2 v_1^3+13 x^4 y^3 v_1^3+13 x^3 y^4 v_1^3+6 x^2 y^5 v_1^3+x y^6 v_1^3\right) d^7+\left(x^8 y \left(\frac{189 v_1^4}{839680}+\frac{3 v_2}{6560}\right)+x y^8 \left(\frac{189 v_1^4}{839680}+\frac{3 v_2}{6560}\right)+x^7 y^2 \left(\frac{993 v_1^4}{419840}+\frac{3 v_2}{1640}\right)+x^2 y^7 \left(\frac{993 v_1^4}{419840}+\frac{3 v_2}{1640}\right)+x^6 y^3 \left(\frac{7299 v_1^4}{839680}+\frac{7 v_2}{1640}\right)+x^3 y^6 \left(\frac{7299 v_1^4}{839680}+\frac{7 v_2}{1640}\right)+x^5 y^4 \left(\frac{6653 v_1^4}{419840}+\frac{21 v_2}{3280}\right)+x^4 y^5 \left(\frac{6653 v_1^4}{419840}+\frac{21 v_2}{3280}\right)\right) d^9+\left(x^{10} y \left(\frac{173 v_1^5}{6717440}+\frac{3 v_1 v_2}{26240}\right)+x y^{10} \left(\frac{173 v_1^5}{6717440}+\frac{3 v_1 v_2}{26240}\right)+x^9 y^2 \left(\frac{567 v_1^5}{1343488}+\frac{9 v_1 v_2}{10496}\right)+x^2 y^9 \left(\frac{567 v_1^5}{1343488}+\frac{9 v_1 v_2}{10496}\right)+x^8 y^3 \left(\frac{3951 v_1^5}{1679360}+\frac{163 v_1 v_2}{52480}\right)+x^3 y^8 \left(\frac{3951 v_1^5}{1679360}+\frac{163 v_1 v_2}{52480}\right)+x^7 y^4 \left(\frac{5567 v_1^5}{839680}+\frac{181 v_1 v_2}{26240}\right)+x^4 y^7 \left(\frac{5567 v_1^5}{839680}+\frac{181 v_1 v_2}{26240}\right)+x^6 y^5 \left(\frac{18219 v_1^5}{1679360}+\frac{133 v_1 v_2}{13120}\right)+x^5 y^6 \left(\frac{18219 v_1^5}{1679360}+\frac{133 v_1 v_2}{13120}\right)\right) d^{11}+\left(x^{12} y \left(\frac{157 v_1^6}{53739520}+\frac{9 v_1^2 v_2}{419840}\right)+x y^{12} \left(\frac{157 v_1^6}{53739520}+\frac{9 v_1^2 v_2}{419840}\right)+x^{11} y^2 \left(\frac{3729 v_1^6}{53739520}+\frac{27 v_1^2 v_2}{104960}\right)+x^2 y^{11} \left(\frac{3729 v_1^6}{53739520}+\frac{27 v_1^2 v_2}{104960}\right)+x^{10} y^3 \left(\frac{29429 v_1^6}{53739520}+\frac{71 v_1^2 v_2}{52480}\right)+x^3 y^{10} \left(\frac{29429 v_1^6}{53739520}+\frac{71 v_1^2 v_2}{52480}\right)+x^9 y^4 \left(\frac{5883 v_1^6}{2686976}+\frac{177 v_1^2 v_2}{41984}\right)+x^4 y^9 \left(\frac{5883 v_1^6}{2686976}+\frac{177 v_1^2 v_2}{41984}\right)+x^8 y^5 \left(\frac{279561 v_1^6}{53739520}+\frac{3637 v_1^2 v_2}{419840}\right)+x^5 y^8 \left(\frac{279561 v_1^6}{53739520}+\frac{3637 v_1^2 v_2}{419840}\right)+x^7 y^6 \left(\frac{53069 v_1^6}{6717440}+\frac{1291 v_1^2 v_2}{104960}\right)+x^6 y^7 \left(\frac{53069 v_1^6}{6717440}+\frac{1291 v_1^2 v_2}{104960}\right)\right) d^{13}+\left(x^{14} y \left(\frac{141 v_1^7}{429916160}+\frac{3 v_1^3 v_2}{839680}\right)+x y^{14} \left(\frac{141 v_1^7}{429916160}+\frac{3 v_1^3 v_2}{839680}\right)+x^{13} y^2 \left(\frac{231 v_1^7}{21495808}+\frac{21 v_1^3 v_2}{335872}\right)+x^2 y^{13} \left(\frac{231 v_1^7}{21495808}+\frac{21 v_1^3 v_2}{335872}\right)+x^{12} y^3 \left(\frac{1201 v_1^7}{10485760}+\frac{37 v_1^3 v_2}{81920}\right)+x^3 y^{12} \left(\frac{1201 v_1^7}{10485760}+\frac{37 v_1^3 v_2}{81920}\right)+x^{11} y^4 \left(\frac{517 v_1^7}{839680}+\frac{6333 v_1^3 v_2}{3358720}\right)+x^4 y^{11} \left(\frac{517 v_1^7}{839680}+\frac{6333 v_1^3 v_2}{3358720}\right)+x^{10} y^5 \left(\frac{85361 v_1^7}{42991616}+\frac{1735 v_1^3 v_2}{335872}\right)+x^5 y^{10} \left(\frac{85361 v_1^7}{42991616}+\frac{1735 v_1^3 v_2}{335872}\right)+x^9 y^6 \left(\frac{1798171 v_1^7}{429916160}+\frac{33137 v_1^3 v_2}{3358720}\right)+x^6 y^9 \left(\frac{1798171 v_1^7}{429916160}+\frac{33137 v_1^3 v_2}{3358720}\right)+x^8 y^7 \left(\frac{2585209 v_1^7}{429916160}+\frac{45493 v_1^3 v_2}{3358720}\right)+x^7 y^8 \left(\frac{2585209 v_1^7}{429916160}+\frac{45493 v_1^3 v_2}{3358720}\right)\right) d^{15}+\left(x^{16} y \left(\frac{25881 v_1^8}{705062502400}+\frac{2979 v_1^4 v_2}{5508300800}+\frac{9 v_2^2}{43033600}\right)+x y^{16} \left(\frac{25881 v_1^8}{705062502400}+\frac{2979 v_1^4 v_2}{5508300800}+\frac{9 v_2^2}{43033600}\right)+x^{15} y^2 \left(\frac{280593 v_1^8}{176265625600}+\frac{9081 v_1^4 v_2}{688537600}+\frac{27 v_2^2}{10758400}\right)+x^2 y^{15} \left(\frac{280593 v_1^8}{176265625600}+\frac{9081 v_1^4 v_2}{688537600}+\frac{27 v_2^2}{10758400}\right)+x^{14} y^3 \left(\frac{15589481 v_1^8}{705062502400}+\frac{347837 v_1^4 v_2}{2754150400}+\frac{39 v_2^2}{2689600}\right)+x^3 y^{14} \left(\frac{15589481 v_1^8}{705062502400}+\frac{347837 v_1^4 v_2}{2754150400}+\frac{39 v_2^2}{2689600}\right)+x^{13} y^4 \left(\frac{678209 v_1^8}{4406640640}+\frac{47047 v_1^4 v_2}{68853760}+\frac{231 v_2^2}{4303360}\right)+x^4 y^{13} \left(\frac{678209 v_1^8}{4406640640}+\frac{47047 v_1^4 v_2}{68853760}+\frac{231 v_2^2}{4303360}\right)+x^{12} y^5 \left(\frac{227224819 v_1^8}{352531251200}+\frac{6634921 v_1^4 v_2}{2754150400}+\frac{1533 v_2^2}{10758400}\right)+x^5 y^{12} \left(\frac{227224819 v_1^8}{352531251200}+\frac{6634921 v_1^4 v_2}{2754150400}+\frac{1533 v_2^2}{10758400}\right)+x^{11} y^6 \left(\frac{1257218737 v_1^8}{705062502400}+\frac{4091531 v_1^4 v_2}{688537600}+\frac{3087 v_2^2}{10758400}\right)+x^6 y^{11} \left(\frac{1257218737 v_1^8}{705062502400}+\frac{4091531 v_1^4 v_2}{688537600}+\frac{3087 v_2^2}{10758400}\right)+x^{10} y^7 \left(\frac{242029301 v_1^8}{70506250240}+\frac{5872809 v_1^4 v_2}{550830080}+\frac{243 v_2^2}{537920}\right)+x^7 y^{10} \left(\frac{242029301 v_1^8}{70506250240}+\frac{5872809 v_1^4 v_2}{550830080}+\frac{243 v_2^2}{537920}\right)+x^9 y^8 \left(\frac{3336593521 v_1^8}{705062502400}+\frac{78290759 v_1^4 v_2}{5508300800}+\frac{24309 v_2^2}{43033600}\right)+x^8 y^9 \left(\frac{3336593521 v_1^8}{705062502400}+\frac{78290759 v_1^4 v_2}{5508300800}+\frac{24309 v_2^2}{43033600}\right)\right) d^{17}+\left(x^{18} y \left(\frac{23113 v_1^9}{5640500019200}+\frac{1701 v_1^5 v_2}{22033203200}+\frac{27 v_1 v_2^2}{344268800}\right)+x y^{18} \left(\frac{23113 v_1^9}{5640500019200}+\frac{1701 v_1^5 v_2}{22033203200}+\frac{27 v_1 v_2^2}{344268800}\right)+x^{17} y^2 \left(\frac{1285029 v_1^9}{5640500019200}+\frac{111051 v_1^5 v_2}{44066406400}+\frac{243 v_1 v_2^2}{172134400}\right)+x^2 y^{17} \left(\frac{1285029 v_1^9}{5640500019200}+\frac{111051 v_1^5 v_2}{44066406400}+\frac{243 v_1 v_2^2}{172134400}\right)+x^{16} y^3 \left(\frac{1413653 v_1^9}{352531251200}+\frac{1369767 v_1^5 v_2}{44066406400}+\frac{3867 v_1 v_2^2}{344268800}\right)+x^3 y^{16} \left(\frac{1413653 v_1^9}{352531251200}+\frac{1369767 v_1^5 v_2}{44066406400}+\frac{3867 v_1 v_2^2}{344268800}\right)+x^{15} y^4 \left(\frac{49542783 v_1^9}{1410125004800}+\frac{1169841 v_1^5 v_2}{5508300800}+\frac{4707 v_1 v_2^2}{86067200}\right)+x^4 y^{15} \left(\frac{49542783 v_1^9}{1410125004800}+\frac{1169841 v_1^5 v_2}{5508300800}+\frac{4707 v_1 v_2^2}{86067200}\right)+x^{14} y^5 \left(\frac{522214951 v_1^9}{2820250009600}+\frac{20669099 v_1^5 v_2}{22033203200}+\frac{3993 v_1 v_2^2}{21516800}\right)+x^5 y^{14} \left(\frac{522214951 v_1^9}{2820250009600}+\frac{20669099 v_1^5 v_2}{22033203200}+\frac{3993 v_1 v_2^2}{21516800}\right)+x^{13} y^6 \left(\frac{365668037 v_1^9}{564050001920}+\frac{3194373 v_1^5 v_2}{1101660160}+\frac{4039 v_1 v_2^2}{8606720}\right)+x^6 y^{13} \left(\frac{365668037 v_1^9}{564050001920}+\frac{3194373 v_1^5 v_2}{1101660160}+\frac{4039 v_1 v_2^2}{8606720}\right)+x^{12} y^7 \left(\frac{1126129163 v_1^9}{705062502400}+\frac{144581173 v_1^5 v_2}{22033203200}+\frac{79069 v_1 v_2^2}{86067200}\right)+x^7 y^{12} \left(\frac{1126129163 v_1^9}{705062502400}+\frac{144581173 v_1^5 v_2}{22033203200}+\frac{79069 v_1 v_2^2}{86067200}\right)+x^{11} y^8 \left(\frac{4042054039 v_1^9}{1410125004800}+\frac{492174319 v_1^5 v_2}{44066406400}+\frac{245007 v_1 v_2^2}{172134400}\right)+x^8 y^{11} \left(\frac{4042054039 v_1^9}{1410125004800}+\frac{492174319 v_1^5 v_2}{44066406400}+\frac{245007 v_1 v_2^2}{172134400}\right)+x^{10} y^9 \left(\frac{10780550281 v_1^9}{2820250009600}+\frac{639838853 v_1^5 v_2}{44066406400}+\frac{304129 v_1 v_2^2}{172134400}\right)+x^9 y^{10} \left(\frac{10780550281 v_1^9}{2820250009600}+\frac{639838853 v_1^5 v_2}{44066406400}+\frac{304129 v_1 v_2^2}{172134400}\right)\right) d^{19}+\left(x^{20} y \left(\frac{20601 v_1^{10}}{45124000153600}+\frac{3729 v_1^6 v_2}{352531251200}+\frac{27 v_1^2 v_2^2}{1377075200}\right)+x y^{20} \left(\frac{20601 v_1^{10}}{45124000153600}+\frac{3729 v_1^6 v_2}{352531251200}+\frac{27 v_1^2 v_2^2}{1377075200}\right)+x^{19} y^2 \left(\frac{57255 v_1^{10}}{1804960006144}+\frac{789 v_1^6 v_2}{1762656256}+\frac{27 v_1^2 v_2^2}{55083008}\right)+x^2 y^{19} \left(\frac{57255 v_1^{10}}{1804960006144}+\frac{789 v_1^6 v_2}{1762656256}+\frac{27 v_1^2 v_2^2}{55083008}\right)+x^{18} y^3 \left(\frac{31212959 v_1^{10}}{45124000153600}+\frac{1224183 v_1^6 v_2}{176265625600}+\frac{14121 v_1^2 v_2^2}{2754150400}\right)+x^3 y^{18} \left(\frac{31212959 v_1^{10}}{45124000153600}+\frac{1224183 v_1^6 v_2}{176265625600}+\frac{14121 v_1^2 v_2^2}{2754150400}\right)+x^{17} y^4 \left(\frac{336642879 v_1^{10}}{45124000153600}+\frac{10342863 v_1^6 v_2}{176265625600}+\frac{87561 v_1^2 v_2^2}{2754150400}\right)+x^4 y^{17} \left(\frac{336642879 v_1^{10}}{45124000153600}+\frac{10342863 v_1^6 v_2}{176265625600}+\frac{87561 v_1^2 v_2^2}{2754150400}\right)+x^{16} y^5 \left(\frac{136260851 v_1^{10}}{2820250009600}+\frac{14009073 v_1^6 v_2}{44066406400}+\frac{185427 v_1^2 v_2^2}{1377075200}\right)+x^5 y^{16} \left(\frac{136260851 v_1^{10}}{2820250009600}+\frac{14009073 v_1^6 v_2}{44066406400}+\frac{185427 v_1^2 v_2^2}{1377075200}\right)+x^{15} y^6 \left(\frac{4701683993 v_1^{10}}{22562000076800}+\frac{53021683 v_1^6 v_2}{44066406400}+\frac{288921 v_1^2 v_2^2}{688537600}\right)+x^6 y^{15} \left(\frac{4701683993 v_1^{10}}{22562000076800}+\frac{53021683 v_1^6 v_2}{44066406400}+\frac{288921 v_1^2 v_2^2}{688537600}\right)+x^{14} y^7 \left(\frac{14363831151 v_1^{10}}{22562000076800}+\frac{589621049 v_1^6 v_2}{176265625600}+\frac{172993 v_1^2 v_2^2}{172134400}\right)+x^7 y^{14} \left(\frac{14363831151 v_1^{10}}{22562000076800}+\frac{589621049 v_1^6 v_2}{176265625600}+\frac{172993 v_1^2 v_2^2}{172134400}\right)+x^{13} y^8 \left(\frac{1615234997 v_1^{10}}{1128100003840}+\frac{496857327 v_1^6 v_2}{70506250240}+\frac{209317 v_1^2 v_2^2}{110166016}\right)+x^8 y^{13} \left(\frac{1615234997 v_1^{10}}{1128100003840}+\frac{496857327 v_1^6 v_2}{70506250240}+\frac{209317 v_1^2 v_2^2}{110166016}\right)+x^{12} y^9 \left(\frac{856265899 v_1^{10}}{352531251200}+\frac{505501721 v_1^6 v_2}{44066406400}+\frac{7938873 v_1^2 v_2^2}{2754150400}\right)+x^9 y^{12} \left(\frac{856265899 v_1^{10}}{352531251200}+\frac{505501721 v_1^6 v_2}{44066406400}+\frac{7938873 v_1^2 v_2^2}{2754150400}\right)+x^{11} y^{10} \left(\frac{71136117417 v_1^{10}}{22562000076800}+\frac{2572869683 v_1^6 v_2}{176265625600}+\frac{4878523 v_1^2 v_2^2}{1377075200}\right)+x^{10} y^{11} \left(\frac{71136117417 v_1^{10}}{22562000076800}+\frac{2572869683 v_1^6 v_2}{176265625600}+\frac{4878523 v_1^2 v_2^2}{1377075200}\right)\right) d^{21}+O[d]^{22}$
$(x+y) d+\frac{1}{624} \left(x^4 y v_1+2 x^3 y^2 v_1+2 x^2 y^3 v_1+x y^4 v_1\right) d^5+\frac{\left(x^8 y v_1^2+6 x^7 y^2 v_1^2+16 x^6 y^3 v_1^2+25 x^5 y^4 v_1^2+25 x^4 y^5 v_1^2+16 x^3 y^6 v_1^2+6 x^2 y^7 v_1^2+x y^8 v_1^2\right) d^9}{389376}+\frac{\left(x^{12} y v_1^3+12 x^{11} y^2 v_1^3+60 x^{10} y^3 v_1^3+175 x^9 y^4 v_1^3+340 x^8 y^5 v_1^3+468 x^7 y^6 v_1^3+468 x^6 y^7 v_1^3+340 x^5 y^8 v_1^3+175 x^4 y^9 v_1^3+60 x^3 y^{10} v_1^3+12 x^2 y^{11} v_1^3+x y^{12} v_1^3\right) d^{13}}{242970624}+\frac{\left(x^{16} y v_1^4+20 x^{15} y^2 v_1^4+160 x^{14} y^3 v_1^4+735 x^{13} y^4 v_1^4+2251 x^{12} y^5 v_1^4+4968 x^{11} y^6 v_1^4+8256 x^{10} y^7 v_1^4+10585 x^9 y^8 v_1^4+10585 x^8 y^9 v_1^4+8256 x^7 y^{10} v_1^4+4968 x^6 y^{11} v_1^4+2251 x^5 y^{12} v_1^4+735 x^4 y^{13} v_1^4+160 x^3 y^{14} v_1^4+20 x^2 y^{15} v_1^4+x y^{16} v_1^4\right) d^{17}}{151613669376}+\frac{\left(x^{20} y v_1^5+30 x^{19} y^2 v_1^5+350 x^{18} y^3 v_1^5+2310 x^{17} y^4 v_1^5+10105 x^{16} y^5 v_1^5+31912 x^{15} y^6 v_1^5+76596 x^{14} y^7 v_1^5+144348 x^{13} y^8 v_1^5+218026 x^{12} y^9 v_1^5+267186 x^{11} y^{10} v_1^5+267186 x^{10} y^{11} v_1^5+218026 x^9 y^{12} v_1^5+144348 x^8 y^{13} v_1^5+76596 x^7 y^{14} v_1^5+31912 x^6 y^{15} v_1^5+10105 x^5 y^{16} v_1^5+2310 x^4 y^{17} v_1^5+350 x^3 y^{18} v_1^5+30 x^2 y^{19} v_1^5+x y^{20} v_1^5\right) d^{21}}{94606929690624}+\left(x^{24} y \left(\frac{15894775065625 v_1^6}{939835406249999984232178384896}+\frac{5 v_2}{59604644775390624}\right)+x y^{24} \left(\frac{15894775065625 v_1^6}{939835406249999984232178384896}+\frac{5 v_2}{59604644775390624}\right)+x^{23} y^2 \left(\frac{111389770183855 v_1^6}{156639234374999997372029730816}+\frac{5 v_2}{4967053731282552}\right)+x^2 y^{23} \left(\frac{111389770183855 v_1^6}{156639234374999997372029730816}+\frac{5 v_2}{4967053731282552}\right)+x^{22} y^3 \left(\frac{111416092001455 v_1^6}{9789952148437499835751858176}+\frac{115 v_2}{14901161193847656}\right)+x^3 y^{22} \left(\frac{111416092001455 v_1^6}{9789952148437499835751858176}+\frac{115 v_2}{14901161193847656}\right)+x^{21} y^4 \left(\frac{1225679464629125 v_1^6}{12049171874999999797848440832}+\frac{1265 v_2}{29802322387695312}\right)+x^4 y^{21} \left(\frac{1225679464629125 v_1^6}{12049171874999999797848440832}+\frac{1265 v_2}{29802322387695312}\right)+x^{20} y^5 \left(\frac{80343519619830865 v_1^6}{134262200892857140604596912128}+\frac{253 v_2}{1419158208937872}\right)+x^5 y^{20} \left(\frac{80343519619830865 v_1^6}{134262200892857140604596912128}+\frac{253 v_2}{1419158208937872}\right)+x^{19} y^6 \left(\frac{28365113120394023 v_1^6}{11188516741071428383716409344}+\frac{1265 v_2}{2128737313406808}\right)+x^6 y^{19} \left(\frac{28365113120394023 v_1^6}{11188516741071428383716409344}+\frac{1265 v_2}{2128737313406808}\right)+x^{18} y^7 \left(\frac{426964507557713725 v_1^6}{52213078124999999124009910272}+\frac{24035 v_2}{14901161193847656}\right)+x^7 y^{18} \left(\frac{426964507557713725 v_1^6}{52213078124999999124009910272}+\frac{24035 v_2}{14901161193847656}\right)+x^{17} y^8 \left(\frac{6525850678086601961 v_1^6}{313278468749999994744059461632}+\frac{24035 v_2}{6622738308376736}\right)+x^8 y^{17} \left(\frac{6525850678086601961 v_1^6}{313278468749999994744059461632}+\frac{24035 v_2}{6622738308376736}\right)+x^{16} y^9 \left(\frac{40381339565896022695 v_1^6}{939835406249999984232178384896}+\frac{408595 v_2}{59604644775390624}\right)+x^9 y^{16} \left(\frac{40381339565896022695 v_1^6}{939835406249999984232178384896}+\frac{408595 v_2}{59604644775390624}\right)+x^{15} y^{10} \left(\frac{879568729652135855 v_1^6}{12049171874999999797848440832}+\frac{81719 v_2}{7450580596923828}\right)+x^{10} y^{15} \left(\frac{879568729652135855 v_1^6}{12049171874999999797848440832}+\frac{81719 v_2}{7450580596923828}\right)+x^{14} y^{11} \left(\frac{2697637749229768841 v_1^6}{26106539062499999562004955136}+\frac{37145 v_2}{2483526865641276}\right)+x^{11} y^{14} \left(\frac{2697637749229768841 v_1^6}{26106539062499999562004955136}+\frac{37145 v_2}{2483526865641276}\right)+x^{13} y^{12} \left(\frac{2746407666909999173 v_1^6}{22377033482142856767432818688}+\frac{37145 v_2}{2128737313406808}\right)+x^{12} y^{13} \left(\frac{2746407666909999173 v_1^6}{22377033482142856767432818688}+\frac{37145 v_2}{2128737313406808}\right)\right) d^{25}+\left(x^{28} y \left(\frac{15869506120729 v_1^7}{586457293499999990160879312175104}+\frac{5 v_1 v_2}{18596649169921874688}\right)+x y^{28} \left(\frac{15869506120729 v_1^7}{586457293499999990160879312175104}+\frac{5 v_1 v_2}{18596649169921874688}\right)+x^{27} y^2 \left(\frac{15901092301849 v_1^7}{10472451669642856967158559145984}+\frac{5 v_1 v_2}{885554722377232128}\right)+x^2 y^{27} \left(\frac{15901092301849 v_1^7}{10472451669642856967158559145984}+\frac{5 v_1 v_2}{885554722377232128}\right)+x^{26} y^3 \left(\frac{779585200599241 v_1^7}{24435720562499999590036638007296}+\frac{295 v_1 v_2}{4649162292480468672}\right)+x^3 y^{26} \left(\frac{779585200599241 v_1^7}{24435720562499999590036638007296}+\frac{295 v_1 v_2}{4649162292480468672}\right)+x^{25} y^4 \left(\frac{2784849541866775 v_1^7}{7518683249999999873857427079168}+\frac{1375 v_1 v_2}{2861022949218749952}\right)+x^4 y^{25} \left(\frac{2784849541866775 v_1^7}{7518683249999999873857427079168}+\frac{1375 v_1 v_2}{2861022949218749952}\right)+x^{24} y^5 \left(\frac{1648495958666858305 v_1^7}{586457293499999990160879312175104}+\frac{100001 v_1 v_2}{37193298339843749376}\right)+x^5 y^{24} \left(\frac{1648495958666858305 v_1^7}{586457293499999990160879312175104}+\frac{100001 v_1 v_2}{37193298339843749376}\right)+x^{23} y^6 \left(\frac{748049146473355721 v_1^7}{48871441124999999180073276014592}+\frac{108851 v_1 v_2}{9298324584960937344}\right)+x^6 y^{23} \left(\frac{748049146473355721 v_1^7}{48871441124999999180073276014592}+\frac{108851 v_1 v_2}{9298324584960937344}\right)+x^{22} y^7 \left(\frac{344237725648985101 v_1^7}{5430160124999999908897030668288}+\frac{127213 v_1 v_2}{3099441528320312448}\right)+x^7 y^{22} \left(\frac{344237725648985101 v_1^7}{5430160124999999908897030668288}+\frac{127213 v_1 v_2}{3099441528320312448}\right)+x^{21} y^8 \left(\frac{390342124050384605 v_1^7}{1879670812499999968464356769792}+\frac{4413079 v_1 v_2}{37193298339843749376}\right)+x^8 y^{21} \left(\frac{390342124050384605 v_1^7}{1879670812499999968464356769792}+\frac{4413079 v_1 v_2}{37193298339843749376}\right)+x^{20} y^9 \left(\frac{324327332212013514385 v_1^7}{586457293499999990160879312175104}+\frac{2674969 v_1 v_2}{9298324584960937344}\right)+x^9 y^{20} \left(\frac{324327332212013514385 v_1^7}{586457293499999990160879312175104}+\frac{2674969 v_1 v_2}{9298324584960937344}\right)+x^{19} y^{10} \left(\frac{119356036010369332225 v_1^7}{97742882249999998360146552029184}+\frac{612007 v_1 v_2}{1033147176106770816}\right)+x^{10} y^{19} \left(\frac{119356036010369332225 v_1^7}{97742882249999998360146552029184}+\frac{612007 v_1 v_2}{1033147176106770816}\right)+x^{18} y^{11} \left(\frac{3518417973385508291 v_1^7}{1551474321428571402542008762368}+\frac{694393 v_1 v_2}{664166041782924096}\right)+x^{11} y^{18} \left(\frac{3518417973385508291 v_1^7}{1551474321428571402542008762368}+\frac{694393 v_1 v_2}{664166041782924096}\right)+x^{17} y^{12} \left(\frac{87448502967322703987 v_1^7}{24435720562499999590036638007296}+\frac{29612431 v_1 v_2}{18596649169921874688}\right)+x^{12} y^{17} \left(\frac{87448502967322703987 v_1^7}{24435720562499999590036638007296}+\frac{29612431 v_1 v_2}{18596649169921874688}\right)+x^{16} y^{13} \left(\frac{944761779800464258337 v_1^7}{195485764499999996720293104058368}+\frac{26068361 v_1 v_2}{12397766113281249792}\right)+x^{13} y^{16} \left(\frac{944761779800464258337 v_1^7}{195485764499999996720293104058368}+\frac{26068361 v_1 v_2}{12397766113281249792}\right)+x^{15} y^{14} \left(\frac{822537025410352705945 v_1^7}{146614323374999997540219828043776}+\frac{11225219 v_1 v_2}{4649162292480468672}\right)+x^{14} y^{15} \left(\frac{822537025410352705945 v_1^7}{146614323374999997540219828043776}+\frac{11225219 v_1 v_2}{4649162292480468672}\right)\right) d^{29}+\left(x^{32} y \left(\frac{15844237175833 v_1^8}{365949351143999993860388690797264896}+\frac{5 v_1^2 v_2}{7736206054687499870208}\right)+x y^{32} \left(\frac{15844237175833 v_1^8}{365949351143999993860388690797264896}+\frac{5 v_1^2 v_2}{7736206054687499870208}\right)+x^{31} y^2 \left(\frac{15886352083993 v_1^8}{5082629876999999914727620705517568}+\frac{5 v_1^2 v_2}{241756439208984370944}\right)+x^2 y^{31} \left(\frac{15886352083993 v_1^8}{5082629876999999914727620705517568}+\frac{5 v_1^2 v_2}{241756439208984370944}\right)+x^{30} y^3 \left(\frac{79502829327485 v_1^8}{952993101937499984011428882284544}+\frac{925 v_1^2 v_2}{2901077270507812451328}\right)+x^3 y^{30} \left(\frac{79502829327485 v_1^8}{952993101937499984011428882284544}+\frac{925 v_1^2 v_2}{2901077270507812451328}\right)+x^{29} y^4 \left(\frac{10623470976312925 v_1^8}{8713079789142856996675921209458688}+\frac{875 v_1^2 v_2}{276293073381696423936}\right)+x^4 y^{29} \left(\frac{10623470976312925 v_1^8}{8713079789142856996675921209458688}+\frac{875 v_1^2 v_2}{276293073381696423936}\right)+x^{28} y^5 \left(\frac{4236372856773064435 v_1^8}{365949351143999993860388690797264896}+\frac{263213 v_1^2 v_2}{11604309082031249805312}\right)+x^5 y^{28} \left(\frac{4236372856773064435 v_1^8}{365949351143999993860388690797264896}+\frac{263213 v_1^2 v_2}{11604309082031249805312}\right)+x^{27} y^6 \left(\frac{256662993696037687 v_1^8}{3267404920928571373753470453547008}+\frac{103289 v_1^2 v_2}{828879220145089271808}\right)+x^6 y^{27} \left(\frac{256662993696037687 v_1^8}{3267404920928571373753470453547008}+\frac{103289 v_1^2 v_2}{828879220145089271808}\right)+x^{26} y^7 \left(\frac{456952632281567293 v_1^8}{1129473305999999981050582379003904}+\frac{1056749 v_1^2 v_2}{1934051513671874967552}\right)+x^7 y^{26} \left(\frac{456952632281567293 v_1^8}{1129473305999999981050582379003904}+\frac{1056749 v_1^2 v_2}{1934051513671874967552}\right)+x^{25} y^8 \left(\frac{15458898892956273605 v_1^8}{9383316695999999842574068994801664}+\frac{3509125 v_1^2 v_2}{1785278320312499970048}\right)+x^8 y^{25} \left(\frac{15458898892956273605 v_1^8}{9383316695999999842574068994801664}+\frac{3509125 v_1^2 v_2}{1785278320312499970048}\right)+x^{24} y^9 \left(\frac{32232870885186107855 v_1^8}{5902408889419354739683688561246208}+\frac{738541 v_1^2 v_2}{124777517011088707584}\right)+x^9 y^{24} \left(\frac{32232870885186107855 v_1^8}{5902408889419354739683688561246208}+\frac{738541 v_1^2 v_2}{124777517011088707584}\right)+x^{23} y^{10} \left(\frac{918071835579518063707 v_1^8}{60991558523999998976731448466210816}+\frac{14644873 v_1^2 v_2}{967025756835937483776}\right)+x^{10} y^{23} \left(\frac{918071835579518063707 v_1^8}{60991558523999998976731448466210816}+\frac{14644873 v_1^2 v_2}{967025756835937483776}\right)+x^{22} y^{11} \left(\frac{356356145653746944663 v_1^8}{10165259753999999829455241411035136}+\frac{64406831 v_1^2 v_2}{1934051513671874967552}\right)+x^{11} y^{22} \left(\frac{356356145653746944663 v_1^8}{10165259753999999829455241411035136}+\frac{64406831 v_1^2 v_2}{1934051513671874967552}\right)+x^{21} y^{12} \left(\frac{27297126725090324993 v_1^8}{390971528999999993440586208116736}+\frac{245562743 v_1^2 v_2}{3868103027343749935104}\right)+x^{12} y^{21} \left(\frac{27297126725090324993 v_1^8}{390971528999999993440586208116736}+\frac{245562743 v_1^2 v_2}{3868103027343749935104}\right)+x^{20} y^{13} \left(\frac{14636936172405523441169 v_1^8}{121983117047999997953462896932421632}+\frac{408524321 v_1^2 v_2}{3868103027343749935104}\right)+x^{13} y^{20} \left(\frac{14636936172405523441169 v_1^8}{121983117047999997953462896932421632}+\frac{408524321 v_1^2 v_2}{3868103027343749935104}\right)+x^{19} y^{14} \left(\frac{195108779150312688715 v_1^8}{1089134973642857124584490151182336}+\frac{42573253 v_1^2 v_2}{276293073381696423936}\right)+x^{14} y^{19} \left(\frac{195108779150312688715 v_1^8}{1089134973642857124584490151182336}+\frac{42573253 v_1^2 v_2}{276293073381696423936}\right)+x^{18} y^{15} \left(\frac{42710669317167842107987 v_1^8}{182974675571999996930194345398632448}+\frac{47830087 v_1^2 v_2}{241756439208984370944}\right)+x^{15} y^{18} \left(\frac{42710669317167842107987 v_1^8}{182974675571999996930194345398632448}+\frac{47830087 v_1^2 v_2}{241756439208984370944}\right)+x^{17} y^{16} \left(\frac{16239514009974475019539 v_1^8}{60991558523999998976731448466210816}+\frac{216671155 v_1^2 v_2}{967025756835937483776}\right)+x^{16} y^{17} \left(\frac{16239514009974475019539 v_1^8}{60991558523999998976731448466210816}+\frac{216671155 v_1^2 v_2}{967025756835937483776}\right)\right) d^{33}+\left(x^{36} y \left(\frac{15818968230937 v_1^9}{228352395113855996168882543057493295104}+\frac{5 v_1^3 v_2}{3620544433593749939257344}\right)+x y^{36} \left(\frac{15818968230937 v_1^9}{228352395113855996168882543057493295104}+\frac{5 v_1^3 v_2}{3620544433593749939257344}\right)+x^{35} y^2 \left(\frac{79347530603645 v_1^9}{12686244172991999787160141280971849728}+\frac{25 v_1^3 v_2}{402282714843749993250816}\right)+x^2 y^{35} \left(\frac{79347530603645 v_1^9}{12686244172991999787160141280971849728}+\frac{25 v_1^3 v_2}{402282714843749993250816}\right)+x^{34} y^3 \left(\frac{41614243167395 v_1^9}{201368955126857139478732401285267456}+\frac{25 v_1^3 v_2}{19893101283482142523392}\right)+x^3 y^{34} \left(\frac{41614243167395 v_1^9}{201368955126857139478732401285267456}+\frac{25 v_1^3 v_2}{19893101283482142523392}\right)+x^{33} y^4 \left(\frac{40347668991919355 v_1^9}{10873923576850285531851549669404442624}+\frac{32725 v_1^3 v_2}{2068882533482142822432768}\right)+x^4 y^{33} \left(\frac{40347668991919355 v_1^9}{10873923576850285531851549669404442624}+\frac{32725 v_1^3 v_2}{2068882533482142822432768}\right)+x^{32} y^5 \left(\frac{4914275593805911199 v_1^9}{114176197556927998084441271528746647552}+\frac{226669 v_1^3 v_2}{1609130859374999973003264}\right)+x^5 y^{32} \left(\frac{4914275593805911199 v_1^9}{114176197556927998084441271528746647552}+\frac{226669 v_1^3 v_2}{1609130859374999973003264}\right)+x^{31} y^6 \left(\frac{70455814785377423 v_1^9}{198222565202999996674377207515185152}+\frac{11957 v_1^3 v_2}{12571334838867187289088}\right)+x^6 y^{31} \left(\frac{70455814785377423 v_1^9}{198222565202999996674377207515185152}+\frac{11957 v_1^3 v_2}{12571334838867187289088}\right)+x^{30} y^7 \left(\frac{14097028121075707139 v_1^9}{6343122086495999893580070640485924864}+\frac{767525 v_1^3 v_2}{150856018066406247469056}\right)+x^7 y^{30} \left(\frac{14097028121075707139 v_1^9}{6343122086495999893580070640485924864}+\frac{767525 v_1^3 v_2}{150856018066406247469056}\right)+x^{29} y^8 \left(\frac{104411723354368428805 v_1^9}{9514683129743999840370105960728887296}+\frac{107300525 v_1^3 v_2}{4827392578124999919009792}\right)+x^8 y^{29} \left(\frac{104411723354368428805 v_1^9}{9514683129743999840370105960728887296}+\frac{107300525 v_1^3 v_2}{4827392578124999919009792}\right)+x^{28} y^9 \left(\frac{629469180424489842205 v_1^9}{14272024694615999760555158941093330944}+\frac{293592769 v_1^3 v_2}{3620544433593749939257344}\right)+x^9 y^{28} \left(\frac{629469180424489842205 v_1^9}{14272024694615999760555158941093330944}+\frac{293592769 v_1^3 v_2}{3620544433593749939257344}\right)+x^{27} y^{10} \left(\frac{25072015216502345267 v_1^9}{169905055888285711435180463584444416}+\frac{5414051 v_1^3 v_2}{21550859723772321067008}\right)+x^{10} y^{27} \left(\frac{25072015216502345267 v_1^9}{169905055888285711435180463584444416}+\frac{5414051 v_1^3 v_2}{21550859723772321067008}\right)+x^{26} y^{11} \left(\frac{2651904477169314586979 v_1^9}{6343122086495999893580070640485924864}+\frac{808001191 v_1^3 v_2}{1206848144531249979752448}\right)+x^{11} y^{26} \left(\frac{2651904477169314586979 v_1^9}{6343122086495999893580070640485924864}+\frac{808001191 v_1^3 v_2}{1206848144531249979752448}\right)+x^{25} y^{12} \left(\frac{212525530298538601459 v_1^9}{209113914939428567920222109027008512}+\frac{82248835 v_1^3 v_2}{53048270089285713395712}\right)+x^{12} y^{25} \left(\frac{212525530298538601459 v_1^9}{209113914939428567920222109027008512}+\frac{82248835 v_1^3 v_2}{53048270089285713395712}\right)+x^{24} y^{13} \left(\frac{20377291863632329702793 v_1^9}{9514683129743999840370105960728887296}+\frac{2846503511 v_1^3 v_2}{905136108398437484814336}\right)+x^{13} y^{24} \left(\frac{20377291863632329702793 v_1^9}{9514683129743999840370105960728887296}+\frac{2846503511 v_1^3 v_2}{905136108398437484814336}\right)+x^{23} y^{14} \left(\frac{12507092558232277646861 v_1^9}{3171561043247999946790035320242962432}+\frac{2261560807 v_1^3 v_2}{402282714843749993250816}\right)+x^{14} y^{23} \left(\frac{12507092558232277646861 v_1^9}{3171561043247999946790035320242962432}+\frac{2261560807 v_1^3 v_2}{402282714843749993250816}\right)+x^{22} y^{15} \left(\frac{121479841347987297520277 v_1^9}{19029366259487999680740211921457774592}+\frac{32222728393 v_1^3 v_2}{3620544433593749939257344}\right)+x^{15} y^{22} \left(\frac{121479841347987297520277 v_1^9}{19029366259487999680740211921457774592}+\frac{32222728393 v_1^3 v_2}{3620544433593749939257344}\right)+x^{21} y^{16} \left(\frac{347369139515296130906831 v_1^9}{38058732518975999361480423842915549184}+\frac{90681200585 v_1^3 v_2}{7241088867187499878514688}\right)+x^{16} y^{21} \left(\frac{347369139515296130906831 v_1^9}{38058732518975999361480423842915549184}+\frac{90681200585 v_1^3 v_2}{7241088867187499878514688}\right)+x^{20} y^{17} \left(\frac{220041125907040850579369 v_1^9}{19029366259487999680740211921457774592}+\frac{4737832537 v_1^3 v_2}{301712036132812494938112}\right)+x^{17} y^{20} \left(\frac{220041125907040850579369 v_1^9}{19029366259487999680740211921457774592}+\frac{4737832537 v_1^3 v_2}{301712036132812494938112}\right)+x^{19} y^{18} \left(\frac{61885692294865179454243 v_1^9}{4757341564871999920185052980364443648}+\frac{63638148713 v_1^3 v_2}{3620544433593749939257344}\right)+x^{18} y^{19} \left(\frac{61885692294865179454243 v_1^9}{4757341564871999920185052980364443648}+\frac{63638148713 v_1^3 v_2}{3620544433593749939257344}\right)\right) d^{37}+O[d]^{38}$
$(x+y) d+\frac{\left(x^6 y v_1+3 x^5 y^2 v_1+5 x^4 y^3 v_1+5 x^3 y^4 v_1+3 x^2 y^5 v_1+x y^6 v_1\right) d^7}{117648}+\frac{\left(x^{12} y v_1^2+9 x^{11} y^2 v_1^2+38 x^{10} y^3 v_1^2+100 x^9 y^4 v_1^2+183 x^8 y^5 v_1^2+245 x^7 y^6 v_1^2+245 x^6 y^7 v_1^2+183 x^5 y^8 v_1^2+100 x^4 y^9 v_1^2+38 x^3 y^{10} v_1^2+9 x^2 y^{11} v_1^2+x y^{12} v_1^2\right) d^{13}}{13841051904}+\frac{\left(x^{18} y v_1^3+18 x^{17} y^2 v_1^3+140 x^{16} y^3 v_1^3+660 x^{15} y^4 v_1^3+2163 x^{14} y^5 v_1^3+5292 x^{13} y^6 v_1^3+10073 x^{12} y^7 v_1^3+15291 x^{11} y^8 v_1^3+18778 x^{10} y^9 v_1^3+18778 x^9 y^{10} v_1^3+15291 x^8 y^{11} v_1^3+10073 x^7 y^{12} v_1^3+5292 x^6 y^{13} v_1^3+2163 x^5 y^{14} v_1^3+660 x^4 y^{15} v_1^3+140 x^3 y^{16} v_1^3+18 x^2 y^{17} v_1^3+x y^{18} v_1^3\right) d^{19}}{1628372074401792}+\frac{\left(x^{24} y v_1^4+30 x^{23} y^2 v_1^4+370 x^{22} y^3 v_1^4+2695 x^{21} y^4 v_1^4+13482 x^{20} y^5 v_1^4+50232 x^{19} y^6 v_1^4+146417 x^{18} y^7 v_1^4+344727 x^{17} y^8 v_1^4+669895 x^{16} y^9 v_1^4+1090386 x^{15} y^{10} v_1^4+1501281 x^{14} y^{11} v_1^4+1759044 x^{13} y^{12} v_1^4+1759044 x^{12} y^{13} v_1^4+1501281 x^{11} y^{14} v_1^4+1090386 x^{10} y^{15} v_1^4+669895 x^9 y^{16} v_1^4+344727 x^8 y^{17} v_1^4+146417 x^7 y^{18} v_1^4+50232 x^6 y^{19} v_1^4+13482 x^5 y^{20} v_1^4+2695 x^4 y^{21} v_1^4+370 x^3 y^{22} v_1^4+30 x^2 y^{23} v_1^4+x y^{24} v_1^4\right) d^{25}}{191574717809222025216}+\frac{\left(x^{30} y v_1^5+45 x^{29} y^2 v_1^5+805 x^{28} y^3 v_1^5+8330 x^{27} y^4 v_1^5+58464 x^{26} y^5 v_1^5+303576 x^{25} y^6 v_1^5+1230617 x^{24} y^7 v_1^5+4036575 x^{23} y^8 v_1^5+10985510 x^{22} y^9 v_1^5+25257694 x^{21} y^{10} v_1^5+49715370 x^{20} y^{11} v_1^5+84595524 x^{19} y^{12} v_1^5+125325240 x^{18} y^{13} v_1^5+162445482 x^{17} y^{14} v_1^5+184804575 x^{16} y^{15} v_1^5+184804575 x^{15} y^{16} v_1^5+162445482 x^{14} y^{17} v_1^5+125325240 x^{13} y^{18} v_1^5+84595524 x^{12} y^{19} v_1^5+49715370 x^{11} y^{20} v_1^5+25257694 x^{10} y^{21} v_1^5+10985510 x^9 y^{22} v_1^5+4036575 x^8 y^{23} v_1^5+1230617 x^7 y^{24} v_1^5+303576 x^6 y^{25} v_1^5+58464 x^5 y^{26} v_1^5+8330 x^4 y^{27} v_1^5+805 x^3 y^{28} v_1^5+45 x^2 y^{29} v_1^5+x y^{30} v_1^5\right) d^{31}}{22538382400819352822611968}+\frac{\left(x^{36} y v_1^6+63 x^{35} y^2 v_1^6+1540 x^{34} y^3 v_1^6+21420 x^{33} y^4 v_1^6+199836 x^{32} y^5 v_1^6+1369368 x^{31} y^6 v_1^6+7294961 x^{30} y^7 v_1^6+31392675 x^{29} y^8 v_1^6+112139540 x^{28} y^9 v_1^6+339246152 x^{27} y^{10} v_1^6+882390024 x^{26} y^{11} v_1^6+1996313904 x^{25} y^{12} v_1^6+3963806640 x^{24} y^{13} v_1^6+6955432950 x^{23} y^{14} v_1^6+10843612350 x^{22} y^{15} v_1^6+15079666480 x^{21} y^{16} v_1^6+18759068100 x^{20} y^{17} v_1^6+20913494940 x^{19} y^{18} v_1^6+20913494940 x^{18} y^{19} v_1^6+18759068100 x^{17} y^{20} v_1^6+15079666480 x^{16} y^{21} v_1^6+10843612350 x^{15} y^{22} v_1^6+6955432950 x^{14} y^{23} v_1^6+3963806640 x^{13} y^{24} v_1^6+1996313904 x^{12} y^{25} v_1^6+882390024 x^{11} y^{26} v_1^6+339246152 x^{10} y^{27} v_1^6+112139540 x^9 y^{28} v_1^6+31392675 x^8 y^{29} v_1^6+7294961 x^7 y^{30} v_1^6+1369368 x^6 y^{31} v_1^6+199836 x^5 y^{32} v_1^6+21420 x^4 y^{33} v_1^6+1540 x^3 y^{34} v_1^6+63 x^2 y^{35} v_1^6+x y^{36} v_1^6\right) d^{37}}{2651595612691595220874652811264}+\frac{\left(x^{42} y v_1^7+84 x^{41} y^2 v_1^7+2688 x^{40} y^3 v_1^7+48300 x^{39} y^4 v_1^7+576576 x^{38} y^5 v_1^7+5021016 x^{37} y^6 v_1^7+33834617 x^{36} y^7 v_1^7+183648447 x^{35} y^8 v_1^7+826327700 x^{34} y^9 v_1^7+3148755096 x^{33} y^{10} v_1^7+10328591052 x^{32} y^{11} v_1^7+29538690480 x^{31} y^{12} v_1^7+74398956984 x^{30} y^{13} v_1^7+166366137285 x^{29} y^{14} v_1^7+332424118596 x^{28} y^{15} v_1^7+596625629828 x^{27} y^{16} v_1^7+965802148056 x^{26} y^{17} v_1^7+1414686478764 x^{25} y^{18} v_1^7+1879716343440 x^{24} y^{19} v_1^7+2269662112260 x^{23} y^{20} v_1^7+2493282220010 x^{22} y^{21} v_1^7+2493282220010 x^{21} y^{22} v_1^7+2269662112260 x^{20} y^{23} v_1^7+1879716343440 x^{19} y^{24} v_1^7+1414686478764 x^{18} y^{25} v_1^7+965802148056 x^{17} y^{26} v_1^7+596625629828 x^{16} y^{27} v_1^7+332424118596 x^{15} y^{28} v_1^7+166366137285 x^{14} y^{29} v_1^7+74398956984 x^{13} y^{30} v_1^7+29538690480 x^{12} y^{31} v_1^7+10328591052 x^{11} y^{32} v_1^7+3148755096 x^{10} y^{33} v_1^7+826327700 x^9 y^{34} v_1^7+183648447 x^8 y^{35} v_1^7+33834617 x^7 y^{36} v_1^7+5021016 x^6 y^{37} v_1^7+576576 x^5 y^{38} v_1^7+48300 x^4 y^{39} v_1^7+2688 x^3 y^{40} v_1^7+84 x^2 y^{41} v_1^7+x y^{42} v_1^7\right) d^{43}}{311954920641940794545461153939587072}+\left(x^{48} y \left(\frac{38996685305289383904141661535767417 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{7 v_2}{36703368217294125441230211032033660188800}\right)+x y^{48} \left(\frac{38996685305289383904141661535767417 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{7 v_2}{36703368217294125441230211032033660188800}\right)+x^{47} y^2 \left(\frac{38996943099307284475899246571457401 v_1^8}{13252079073516112123140453915086711096436166535204935405235706792404582400}+\frac{7 v_2}{1529307009053921893384592126334735841200}\right)+x^2 y^{47} \left(\frac{38996943099307284475899246571457401 v_1^8}{13252079073516112123140453915086711096436166535204935405235706792404582400}+\frac{7 v_2}{1529307009053921893384592126334735841200}\right)+x^{46} y^3 \left(\frac{14233900730064304470295710440866110341 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{329 v_2}{4587921027161765680153776379004207523600}\right)+x^3 y^{46} \left(\frac{14233900730064304470295710440866110341 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{329 v_2}{4587921027161765680153776379004207523600}\right)+x^{45} y^4 \left(\frac{641305701667143814162832720622239579093 v_1^8}{238537423323290018216528170471560799735850997633688837294242722263282483200}+\frac{7567 v_2}{9175842054323531360307552758008415047200}\right)+x^4 y^{45} \left(\frac{641305701667143814162832720622239579093 v_1^8}{238537423323290018216528170471560799735850997633688837294242722263282483200}+\frac{7567 v_2}{9175842054323531360307552758008415047200}\right)+x^{44} y^5 \left(\frac{3173069545689294204853228246290276939679 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{7567 v_2}{1019538006035947928923061417556490560800}\right)+x^5 y^{44} \left(\frac{3173069545689294204853228246290276939679 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{7567 v_2}{1019538006035947928923061417556490560800}\right)+x^{43} y^6 \left(\frac{51220818759067443586774797436975775355319 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{83237 v_2}{1529307009053921893384592126334735841200}\right)+x^6 y^{43} \left(\frac{51220818759067443586774797436975775355319 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{83237 v_2}{1529307009053921893384592126334735841200}\right)+x^{42} y^7 \left(\frac{5095155194564786650517067208410182003521397 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{11891 v_2}{35565279280323764962432375031040368400}\right)+x^7 y^{42} \left(\frac{5095155194564786650517067208410182003521397 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{11891 v_2}{35565279280323764962432375031040368400}\right)+x^{41} y^8 \left(\frac{5651884355228658507587659129764276858710963 v_1^8}{238537423323290018216528170471560799735850997633688837294242722263282483200}+\frac{83237 v_2}{47420372373765019949909833374720491200}\right)+x^8 y^{41} \left(\frac{5651884355228658507587659129764276858710963 v_1^8}{238537423323290018216528170471560799735850997633688837294242722263282483200}+\frac{83237 v_2}{47420372373765019949909833374720491200}\right)+x^{40} y^9 \left(\frac{10372729960103601362288728802849581579939729 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{3412717 v_2}{426783351363885179549188500372484420800}\right)+x^9 y^{40} \left(\frac{10372729960103601362288728802849581579939729 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{3412717 v_2}{426783351363885179549188500372484420800}\right)+x^{39} y^{10} \left(\frac{9058627011112741288235236219295848287878503 v_1^8}{14908588957705626138533010654472549983490687352105552330890170141455155200}+\frac{3412717 v_2}{106695837840971294887297125093121105200}\right)+x^{10} y^{39} \left(\frac{9058627011112741288235236219295848287878503 v_1^8}{14908588957705626138533010654472549983490687352105552330890170141455155200}+\frac{3412717 v_2}{106695837840971294887297125093121105200}\right)+x^{38} y^{11} \left(\frac{1862182046052373290223020123928365010617121 v_1^8}{764543023472083391719641572024233332486701915492592427225136930331033600}+\frac{310247 v_2}{2735790713871058843264028848541566800}\right)+x^{11} y^{38} \left(\frac{1862182046052373290223020123928365010617121 v_1^8}{764543023472083391719641572024233332486701915492592427225136930331033600}+\frac{310247 v_2}{2735790713871058843264028848541566800}\right)+x^{37} y^{12} \left(\frac{2056488659728579158588530515040658652799771 v_1^8}{241434638991184228964097338533968420785274289102923924386885346420326400}+\frac{310247 v_2}{863933909643492266293903846907863200}\right)+x^{12} y^{37} \left(\frac{2056488659728579158588530515040658652799771 v_1^8}{241434638991184228964097338533968420785274289102923924386885346420326400}+\frac{310247 v_2}{863933909643492266293903846907863200}\right)+x^{36} y^{13} \left(\frac{1566577364534883552577765312327769228475946513 v_1^8}{59634355830822504554132042617890199933962749408422209323560680565820620800}+\frac{11479139 v_2}{11231140825365399461820750009802221600}\right)+x^{13} y^{36} \left(\frac{1566577364534883552577765312327769228475946513 v_1^8}{59634355830822504554132042617890199933962749408422209323560680565820620800}+\frac{11479139 v_2}{11231140825365399461820750009802221600}\right)+x^{35} y^{14} \left(\frac{10584240611714575304735046293361554649672629 v_1^8}{146837441257796256212082591856916466442506000390082386761614479694233600}+\frac{1639877 v_2}{623952268075855525656708333877901200}\right)+x^{14} y^{35} \left(\frac{10584240611714575304735046293361554649672629 v_1^8}{146837441257796256212082591856916466442506000390082386761614479694233600}+\frac{1639877 v_2}{623952268075855525656708333877901200}\right)+x^{34} y^{15} \left(\frac{7046228400679869485882580793231327972254824627 v_1^8}{39756237220548336369421361745260133289308499605614806215707120377213747200}+\frac{11479139 v_2}{1871856804227566576970125001633703600}\right)+x^{15} y^{34} \left(\frac{7046228400679869485882580793231327972254824627 v_1^8}{39756237220548336369421361745260133289308499605614806215707120377213747200}+\frac{11479139 v_2}{1871856804227566576970125001633703600}\right)+x^{33} y^{16} \left(\frac{580221481594428871775722013126015764641988387 v_1^8}{1477011909122538812486242541619571515392266239217887537425651531041996800}+\frac{11479139 v_2}{880873790224737212691823530180566400}\right)+x^{16} y^{33} \left(\frac{580221481594428871775722013126015764641988387 v_1^8}{1477011909122538812486242541619571515392266239217887537425651531041996800}+\frac{11479139 v_2}{880873790224737212691823530180566400}\right)+x^{32} y^{17} \left(\frac{125424627597803049675008158611257741053524262483 v_1^8}{159024948882193345477685446981040533157233998422459224862828481508854988800}+\frac{126270529 v_2}{4991618144606844205253666671023209600}\right)+x^{17} y^{32} \left(\frac{125424627597803049675008158611257741053524262483 v_1^8}{159024948882193345477685446981040533157233998422459224862828481508854988800}+\frac{126270529 v_2}{4991618144606844205253666671023209600}\right)+x^{31} y^{18} \left(\frac{210245464403489535927887174762964595776019957 v_1^8}{145983735203971859373640251206585556753886779457581907768814395509964800}+\frac{126270529 v_2}{2807785206341349865455187502450555400}\right)+x^{18} y^{31} \left(\frac{210245464403489535927887174762964595776019957 v_1^8}{145983735203971859373640251206585556753886779457581907768814395509964800}+\frac{126270529 v_2}{2807785206341349865455187502450555400}\right)+x^{30} y^{19} \left(\frac{286209194049904623470811400997051430289830089681 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{3914386399 v_2}{53347918920485647443648562546560552600}\right)+x^{19} y^{30} \left(\frac{286209194049904623470811400997051430289830089681 v_1^8}{119268711661645009108264085235780399867925498816844418647121361131641241600}+\frac{3914386399 v_2}{53347918920485647443648562546560552600}\right)+x^{29} y^{20} \left(\frac{290884641014423709584405494862271007139125526281 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{3914386399 v_2}{35565279280323764962432375031040368400}\right)+x^{20} y^{29} \left(\frac{290884641014423709584405494862271007139125526281 v_1^8}{79512474441096672738842723490520266578616999211229612431414240754427494400}+\frac{3914386399 v_2}{35565279280323764962432375031040368400}\right)+x^{28} y^{21} \left(\frac{7318832325711768715162946661128368893005838028967 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{16216743653 v_2}{106695837840971294887297125093121105200}\right)+x^{21} y^{28} \left(\frac{7318832325711768715162946661128368893005838028967 v_1^8}{1431224539939740109299169022829364798415105985802133023765456333579694899200}+\frac{16216743653 v_2}{106695837840971294887297125093121105200}\right)+x^{27} y^{22} \left(\frac{1174451126461785364861997388204190543954818467051 v_1^8}{178903067492467513662396127853670599801888248225266627970682041697461862400}+\frac{10319745961 v_2}{53347918920485647443648562546560552600}\right)+x^{22} y^{27} \left(\frac{1174451126461785364861997388204190543954818467051 v_1^8}{178903067492467513662396127853670599801888248225266627970682041697461862400}+\frac{10319745961 v_2}{53347918920485647443648562546560552600}\right)+x^{26} y^{23} \left(\frac{154039236574250452919875425906122115685698472361 v_1^8}{19878118610274168184710680872630066644654249802807403107853560188606873600}+\frac{1346053821 v_2}{5927546546720627493738729171840061400}\right)+x^{23} y^{26} \left(\frac{154039236574250452919875425906122115685698472361 v_1^8}{19878118610274168184710680872630066644654249802807403107853560188606873600}+\frac{1346053821 v_2}{5927546546720627493738729171840061400}\right)+x^{25} y^{24} \left(\frac{77227960019885867954684861913163565783186136443 v_1^8}{9174516281665000700635698864290799989840422985911109126701643163972403200}+\frac{448684607 v_2}{1823860475914039228842685899027711200}\right)+x^{24} y^{25} \left(\frac{77227960019885867954684861913163565783186136443 v_1^8}{9174516281665000700635698864290799989840422985911109126701643163972403200}+\frac{448684607 v_2}{1823860475914039228842685899027711200}\right)\right) d^{49}+O[d]^{50}$
$(x+y) d+\frac{\left(x^{10} y v_1+5 x^9 y^2 v_1+15 x^8 y^3 v_1+30 x^7 y^4 v_1+42 x^6 y^5 v_1+42 x^5 y^6 v_1+30 x^4 y^7 v_1+15 x^3 y^8 v_1+5 x^2 y^9 v_1+x y^{10} v_1\right) d^{11}}{25937424600}+\frac{\left(x^{20} y v_1^2+15 x^{19} y^2 v_1^2+110 x^{18} y^3 v_1^2+525 x^{17} y^4 v_1^2+1827 x^{16} y^5 v_1^2+4914 x^{15} y^6 v_1^2+10560 x^{14} y^7 v_1^2+18495 x^{13} y^8 v_1^2+26720 x^{12} y^9 v_1^2+32065 x^{11} y^{10} v_1^2+32065 x^{10} y^{11} v_1^2+26720 x^9 y^{12} v_1^2+18495 x^8 y^{13} v_1^2+10560 x^7 y^{14} v_1^2+4914 x^6 y^{15} v_1^2+1827 x^5 y^{16} v_1^2+525 x^4 y^{17} v_1^2+110 x^3 y^{18} v_1^2+15 x^2 y^{19} v_1^2+x y^{20} v_1^2\right) d^{21}}{672749994880685160000}+\frac{\left(x^{30} y v_1^3+30 x^{29} y^2 v_1^3+400 x^{28} y^3 v_1^3+3325 x^{27} y^4 v_1^3+19782 x^{26} y^5 v_1^3+90636 x^{25} y^6 v_1^3+334260 x^{24} y^7 v_1^3+1021275 x^{23} y^8 v_1^3+2636645 x^{22} y^9 v_1^3+5832684 x^{21} y^{10} v_1^3+11167189 x^{20} y^{11} v_1^3+18638700 x^{19} y^{12} v_1^3+27259650 x^{18} y^{13} v_1^3+35058540 x^{17} y^{14} v_1^3+39737331 x^{16} y^{15} v_1^3+39737331 x^{15} y^{16} v_1^3+35058540 x^{14} y^{17} v_1^3+27259650 x^{13} y^{18} v_1^3+18638700 x^{12} y^{19} v_1^3+11167189 x^{11} y^{20} v_1^3+5832684 x^{10} y^{21} v_1^3+2636645 x^9 y^{22} v_1^3+1021275 x^8 y^{23} v_1^3+334260 x^7 y^{24} v_1^3+90636 x^6 y^{25} v_1^3+19782 x^5 y^{26} v_1^3+3325 x^4 y^{27} v_1^3+400 x^3 y^{28} v_1^3+30 x^2 y^{29} v_1^3+x y^{30} v_1^3\right) d^{31}}{17449402266868157333838936000000}+\frac{\left(x^{40} y v_1^4+50 x^{39} y^2 v_1^4+1050 x^{38} y^3 v_1^4+13300 x^{37} y^4 v_1^4+118202 x^{36} y^5 v_1^4+799848 x^{35} y^6 v_1^4+4333500 x^{34} y^7 v_1^4+19438650 x^{33} y^8 v_1^4+73911695 x^{32} y^9 v_1^4+242350108 x^{31} y^{10} v_1^4+694153857 x^{30} y^{11} v_1^4+1754023340 x^{29} y^{12} v_1^4+3940080880 x^{28} y^{13} v_1^4+7915219500 x^{27} y^{14} v_1^4+14287126446 x^{26} y^{15} v_1^4+23256285660 x^{25} y^{16} v_1^4+34235345340 x^{24} y^{17} v_1^4+45673941090 x^{23} y^{18} v_1^4+55306910055 x^{22} y^{19} v_1^4+60845867940 x^{21} y^{20} v_1^4+60845867940 x^{20} y^{21} v_1^4+55306910055 x^{19} y^{22} v_1^4+45673941090 x^{18} y^{23} v_1^4+34235345340 x^{17} y^{24} v_1^4+23256285660 x^{16} y^{25} v_1^4+14287126446 x^{15} y^{26} v_1^4+7915219500 x^{14} y^{27} v_1^4+3940080880 x^{13} y^{28} v_1^4+1754023340 x^{12} y^{29} v_1^4+694153857 x^{11} y^{30} v_1^4+242350108 x^{10} y^{31} v_1^4+73911695 x^9 y^{32} v_1^4+19438650 x^8 y^{33} v_1^4+4333500 x^7 y^{34} v_1^4+799848 x^6 y^{35} v_1^4+118202 x^5 y^{36} v_1^4+13300 x^4 y^{37} v_1^4+1050 x^3 y^{38} v_1^4+50 x^2 y^{39} v_1^4+x y^{40} v_1^4\right) d^{41}}{452592555611961908987384431044225600000000}+\frac{\left(x^{50} y v_1^5+75 x^{49} y^2 v_1^5+2275 x^{48} y^3 v_1^5+40600 x^{47} y^4 v_1^5+499842 x^{46} y^5 v_1^5+4631970 x^{45} y^6 v_1^5+34110450 x^{44} y^7 v_1^5+207046125 x^{43} y^8 v_1^5+1063132070 x^{42} y^9 v_1^5+4707504802 x^{41} y^{10} v_1^5+18240308119 x^{40} y^{11} v_1^5+62555050400 x^{39} y^{12} v_1^5+191605231930 x^{38} y^{13} v_1^5+527986560460 x^{37} y^{14} v_1^5+1316653942774 x^{36} y^{15} v_1^5+2985727390947 x^{35} y^{16} v_1^5+6181319503485 x^{34} y^{17} v_1^5+11721491484395 x^{33} y^{18} v_1^5+20413653094770 x^{32} y^{19} v_1^5+32722572560860 x^{31} y^{20} v_1^5+48365238083812 x^{30} y^{21} v_1^5+66006957723630 x^{29} y^{22} v_1^5+83269626432760 x^{28} y^{23} v_1^5+97177536089200 x^{27} y^{24} v_1^5+104966446824936 x^{26} y^{25} v_1^5+104966446824936 x^{25} y^{26} v_1^5+97177536089200 x^{24} y^{27} v_1^5+83269626432760 x^{23} y^{28} v_1^5+66006957723630 x^{22} y^{29} v_1^5+48365238083812 x^{21} y^{30} v_1^5+32722572560860 x^{20} y^{31} v_1^5+20413653094770 x^{19} y^{32} v_1^5+11721491484395 x^{18} y^{33} v_1^5+6181319503485 x^{17} y^{34} v_1^5+2985727390947 x^{16} y^{35} v_1^5+1316653942774 x^{15} y^{36} v_1^5+527986560460 x^{14} y^{37} v_1^5+191605231930 x^{13} y^{38} v_1^5+62555050400 x^{12} y^{39} v_1^5+18240308119 x^{11} y^{40} v_1^5+4707504802 x^{10} y^{41} v_1^5+1063132070 x^9 y^{42} v_1^5+207046125 x^8 y^{43} v_1^5+34110450 x^7 y^{44} v_1^5+4631970 x^6 y^{45} v_1^5+499842 x^5 y^{46} v_1^5+40600 x^4 y^{47} v_1^5+2275 x^3 y^{48} v_1^5+75 x^2 y^{49} v_1^5+x y^{50} v_1^5\right) d^{51}}{11739085285706568872432346031423500765389760000000000}+\frac{\left(x^{60} y v_1^6+105 x^{59} y^2 v_1^6+4340 x^{58} y^3 v_1^6+103530 x^{57} y^4 v_1^6+1680084 x^{56} y^5 v_1^6+20312754 x^{55} y^6 v_1^6+193710660 x^{54} y^7 v_1^6+1514593080 x^{53} y^8 v_1^6+9982402430 x^{52} y^9 v_1^6+56615997438 x^{51} y^{10} v_1^6+280732659877 x^{50} y^{11} v_1^6+1232274466550 x^{49} y^{12} v_1^6+4836332067105 x^{48} y^{13} v_1^6+17109696497020 x^{47} y^{14} v_1^6+54927036172890 x^{46} y^{15} v_1^6+160900954950960 x^{45} y^{16} v_1^6+432095599759635 x^{44} y^{17} v_1^6+1067955096404625 x^{43} y^{18} v_1^6+2437364192169270 x^{42} y^{19} v_1^6+5151185143538980 x^{41} y^{20} v_1^6+10105431803674350 x^{40} y^{21} v_1^6+18439486163844050 x^{39} y^{22} v_1^6+31350118354387210 x^{38} y^{23} v_1^6+49734561555585060 x^{37} y^{24} v_1^6+73711336128981344 x^{36} y^{25} v_1^6+102164993412262956 x^{35} y^{26} v_1^6+132529409720182545 x^{34} y^{27} v_1^6+161004332684400190 x^{33} y^{28} v_1^6+183264495556558845 x^{32} y^{29} v_1^6+195508719268445492 x^{31} y^{30} v_1^6+195508719268445492 x^{30} y^{31} v_1^6+183264495556558845 x^{29} y^{32} v_1^6+161004332684400190 x^{28} y^{33} v_1^6+132529409720182545 x^{27} y^{34} v_1^6+102164993412262956 x^{26} y^{35} v_1^6+73711336128981344 x^{25} y^{36} v_1^6+49734561555585060 x^{24} y^{37} v_1^6+31350118354387210 x^{23} y^{38} v_1^6+18439486163844050 x^{22} y^{39} v_1^6+10105431803674350 x^{21} y^{40} v_1^6+5151185143538980 x^{20} y^{41} v_1^6+2437364192169270 x^{19} y^{42} v_1^6+1067955096404625 x^{18} y^{43} v_1^6+432095599759635 x^{17} y^{44} v_1^6+160900954950960 x^{16} y^{45} v_1^6+54927036172890 x^{15} y^{46} v_1^6+17109696497020 x^{14} y^{47} v_1^6+4836332067105 x^{13} y^{48} v_1^6+1232274466550 x^{12} y^{49} v_1^6+280732659877 x^{11} y^{50} v_1^6+56615997438 x^{10} y^{51} v_1^6+9982402430 x^9 y^{52} v_1^6+1514593080 x^8 y^{53} v_1^6+193710660 x^7 y^{54} v_1^6+20312754 x^6 y^{55} v_1^6+1680084 x^5 y^{56} v_1^6+103530 x^4 y^{57} v_1^6+4340 x^3 y^{58} v_1^6+105 x^2 y^{59} v_1^6+x y^{60} v_1^6\right) d^{61}}{304481639470983587853420993791156281770339189612096000000000000}+\frac{\left(x^{70} y v_1^7+140 x^{69} y^2 v_1^7+7560 x^{68} y^3 v_1^7+232050 x^{67} y^4 v_1^7+4789554 x^{66} y^5 v_1^7+72997848 x^{65} y^6 v_1^7+871547820 x^{64} y^7 v_1^7+8486975640 x^{63} y^8 v_1^7+69391231910 x^{62} y^9 v_1^7+486841635280 x^{61} y^{10} v_1^7+2980490819157 x^{60} y^{11} v_1^7+16134728562330 x^{59} y^{12} v_1^7+78063177080280 x^{58} y^{13} v_1^7+340514287240200 x^{57} y^{14} v_1^7+1348881327292236 x^{56} y^{15} v_1^7+4881985594593492 x^{55} y^{16} v_1^7+16226754810667905 x^{54} y^{17} v_1^7+49748218947276360 x^{53} y^{18} v_1^7+141208707557022630 x^{52} y^{19} v_1^7+372293798837551500 x^{51} y^{20} v_1^7+914247377198877072 x^{50} y^{21} v_1^7+2096273796314337675 x^{49} y^{22} v_1^7+4497322102613243085 x^{48} y^{23} v_1^7+9044369094117937020 x^{47} y^{24} v_1^7+17077093066841288544 x^{46} y^{25} v_1^7+30315386317682852190 x^{45} y^{26} v_1^7+50657905104266684595 x^{44} y^{27} v_1^7+79765604774875282270 x^{43} y^{28} v_1^7+118454819090401101560 x^{42} y^{29} v_1^7+166028843135960950698 x^{41} y^{30} v_1^7+219775230331639860990 x^{40} y^{31} v_1^7+274889670620351792145 x^{39} y^{32} v_1^7+325008822945815611575 x^{38} y^{33} v_1^7+363342646099235668600 x^{37} y^{34} v_1^7+384154671476102636984 x^{36} y^{35} v_1^7+384154671476102636984 x^{35} y^{36} v_1^7+363342646099235668600 x^{34} y^{37} v_1^7+325008822945815611575 x^{33} y^{38} v_1^7+274889670620351792145 x^{32} y^{39} v_1^7+219775230331639860990 x^{31} y^{40} v_1^7+166028843135960950698 x^{30} y^{41} v_1^7+118454819090401101560 x^{29} y^{42} v_1^7+79765604774875282270 x^{28} y^{43} v_1^7+50657905104266684595 x^{27} y^{44} v_1^7+30315386317682852190 x^{26} y^{45} v_1^7+17077093066841288544 x^{25} y^{46} v_1^7+9044369094117937020 x^{24} y^{47} v_1^7+4497322102613243085 x^{23} y^{48} v_1^7+2096273796314337675 x^{22} y^{49} v_1^7+914247377198877072 x^{21} y^{50} v_1^7+372293798837551500 x^{20} y^{51} v_1^7+141208707557022630 x^{19} y^{52} v_1^7+49748218947276360 x^{18} y^{53} v_1^7+16226754810667905 x^{17} y^{54} v_1^7+4881985594593492 x^{16} y^{55} v_1^7+1348881327292236 x^{15} y^{56} v_1^7+340514287240200 x^{14} y^{57} v_1^7+78063177080280 x^{13} y^{58} v_1^7+16134728562330 x^{12} y^{59} v_1^7+2980490819157 x^{11} y^{60} v_1^7+486841635280 x^{10} y^{61} v_1^7+69391231910 x^9 y^{62} v_1^7+8486975640 x^8 y^{63} v_1^7+871547820 x^7 y^{64} v_1^7+72997848 x^6 y^{65} v_1^7+4789554 x^5 y^{66} v_1^7+232050 x^4 y^{67} v_1^7+7560 x^3 y^{68} v_1^7+140 x^2 y^{69} v_1^7+x y^{70} v_1^7\right) d^{71}}{7897469565863020697785582878515184205234527246988843247961600000000000000}+O[d]^{72}$
$(x+y) d+\frac{\left(x^{12} y v_1+6 x^{11} y^2 v_1+22 x^{10} y^3 v_1+55 x^9 y^4 v_1+99 x^8 y^5 v_1+132 x^7 y^6 v_1+132 x^6 y^7 v_1+99 x^5 y^8 v_1+55 x^4 y^9 v_1+22 x^3 y^{10} v_1+6 x^2 y^{11} v_1+x y^{12} v_1\right) d^{13}}{23298085122480}+\frac{\left(x^{24} y v_1^2+18 x^{23} y^2 v_1^2+160 x^{22} y^3 v_1^2+935 x^{21} y^4 v_1^2+4026 x^{20} y^5 v_1^2+13552 x^{19} y^6 v_1^2+36916 x^{18} y^7 v_1^2+83160 x^{17} y^8 v_1^2+157135 x^{16} y^9 v_1^2+251438 x^{15} y^{10} v_1^2+342876 x^{14} y^{11} v_1^2+400023 x^{13} y^{12} v_1^2+400023 x^{12} y^{13} v_1^2+342876 x^{11} y^{14} v_1^2+251438 x^{10} y^{15} v_1^2+157135 x^9 y^{16} v_1^2+83160 x^8 y^{17} v_1^2+36916 x^7 y^{18} v_1^2+13552 x^6 y^{19} v_1^2+4026 x^5 y^{20} v_1^2+935 x^4 y^{21} v_1^2+160 x^3 y^{22} v_1^2+18 x^2 y^{23} v_1^2+x y^{24} v_1^2\right) d^{25}}{542800770374323916601350400}+\frac{\left(x^{36} y v_1^3+36 x^{35} y^2 v_1^3+580 x^{34} y^3 v_1^3+5865 x^{33} y^4 v_1^3+42735 x^{32} y^5 v_1^3+241472 x^{31} y^6 v_1^3+1106292 x^{30} y^7 v_1^3+4231755 x^{29} y^8 v_1^3+13792790 x^{28} y^9 v_1^3+38871250 x^{27} y^{10} v_1^3+95754126 x^{26} y^{11} v_1^3+207867296 x^{25} y^{12} v_1^3+400144823 x^{24} y^{13} v_1^3+686305428 x^{23} y^{14} v_1^3+1052586400 x^{22} y^{15} v_1^3+1447463215 x^{21} y^{16} v_1^3+1788124800 x^{20} y^{17} v_1^3+1986837776 x^{19} y^{18} v_1^3+1986837776 x^{18} y^{19} v_1^3+1788124800 x^{17} y^{20} v_1^3+1447463215 x^{16} y^{21} v_1^3+1052586400 x^{15} y^{22} v_1^3+686305428 x^{14} y^{23} v_1^3+400144823 x^{13} y^{24} v_1^3+207867296 x^{12} y^{25} v_1^3+95754126 x^{11} y^{26} v_1^3+38871250 x^{10} y^{27} v_1^3+13792790 x^9 y^{28} v_1^3+4231755 x^8 y^{29} v_1^3+1106292 x^7 y^{30} v_1^3+241472 x^6 y^{31} v_1^3+42735 x^5 y^{32} v_1^3+5865 x^4 y^{33} v_1^3+580 x^3 y^{34} v_1^3+36 x^2 y^{35} v_1^3+x y^{36} v_1^3\right) d^{37}}{12646218552728718781958366039317396992000}+\frac{\left(x^{48} y v_1^4+60 x^{47} y^2 v_1^4+1520 x^{46} y^3 v_1^4+23345 x^{45} y^4 v_1^4+252840 x^{44} y^5 v_1^4+2095632 x^{43} y^6 v_1^4+13979460 x^{42} y^7 v_1^4+77623920 x^{41} y^8 v_1^4+367412870 x^{40} y^9 v_1^4+1508522730 x^{39} y^{10} v_1^4+5444152896 x^{38} y^{11} v_1^4+17447684800 x^{37} y^{12} v_1^4+50058940023 x^{36} y^{13} v_1^4+129409294056 x^{35} y^{14} v_1^4+303007605780 x^{34} y^{15} v_1^4+645338624265 x^{33} y^{16} v_1^4+1254504266400 x^{32} y^{17} v_1^4+2232216568736 x^{31} y^{18} v_1^4+3644024003312 x^{30} y^{19} v_1^4+5467822470330 x^{29} y^{20} v_1^4+7552244078390 x^{28} y^{21} v_1^4+9612982040800 x^{27} y^{22} v_1^4+11285445678378 x^{26} y^{23} v_1^4+12226195896096 x^{25} y^{24} v_1^4+12226195896096 x^{24} y^{25} v_1^4+11285445678378 x^{23} y^{26} v_1^4+9612982040800 x^{22} y^{27} v_1^4+7552244078390 x^{21} y^{28} v_1^4+5467822470330 x^{20} y^{29} v_1^4+3644024003312 x^{19} y^{30} v_1^4+2232216568736 x^{18} y^{31} v_1^4+1254504266400 x^{17} y^{32} v_1^4+645338624265 x^{16} y^{33} v_1^4+303007605780 x^{15} y^{34} v_1^4+129409294056 x^{14} y^{35} v_1^4+50058940023 x^{13} y^{36} v_1^4+17447684800 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$(x+y) d+\frac{\left(x^{16} y v_1+8 x^{15} y^2 v_1+40 x^{14} y^3 v_1+140 x^{13} y^4 v_1+364 x^{12} y^5 v_1+728 x^{11} y^6 v_1+1144 x^{10} y^7 v_1+1430 x^9 y^8 v_1+1430 x^8 y^9 v_1+1144 x^7 y^{10} v_1+728 x^6 y^{11} v_1+364 x^5 y^{12} v_1+140 x^4 y^{13} v_1+40 x^3 y^{14} v_1+8 x^2 y^{15} v_1+x y^{16} v_1\right) d^{17}}{48661191875666868480}+\frac{\left(x^{32} y v_1^2+24 x^{31} y^2 v_1^2+288 x^{30} y^3 v_1^2+2300 x^{29} y^4 v_1^2+13704 x^{28} y^5 v_1^2+64680 x^{27} y^6 v_1^2+250624 x^{26} y^7 v_1^2+815958 x^{25} y^8 v_1^2+2267980 x^{24} y^9 v_1^2+5444296 x^{23} y^{10} v_1^2+11384256 x^{22} y^{11} v_1^2+20871500 x^{21} y^{12} v_1^2+33715640 x^{20} y^{13} v_1^2+48165240 x^{19} y^{14} v_1^2+61009312 x^{18} y^{15} v_1^2+68635477 x^{17} y^{16} v_1^2+68635477 x^{16} y^{17} v_1^2+61009312 x^{15} y^{18} v_1^2+48165240 x^{14} y^{19} v_1^2+33715640 x^{13} y^{20} v_1^2+20871500 x^{12} y^{21} v_1^2+11384256 x^{11} y^{22} v_1^2+5444296 x^{10} y^{23} v_1^2+2267980 x^9 y^{24} v_1^2+815958 x^8 y^{25} 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$(x+y) d+\frac{\left(x^{18} y v_1+9 x^{17} y^2 v_1+51 x^{16} y^3 v_1+204 x^{15} y^4 v_1+612 x^{14} y^5 v_1+1428 x^{13} y^6 v_1+2652 x^{12} y^7 v_1+3978 x^{11} y^8 v_1+4862 x^{10} y^9 v_1+4862 x^9 y^{10} v_1+3978 x^8 y^{11} v_1+2652 x^7 y^{12} v_1+1428 x^6 y^{13} v_1+612 x^5 y^{14} v_1+204 x^4 y^{15} v_1+51 x^3 y^{16} v_1+9 x^2 y^{17} v_1+x y^{18} v_1\right) d^{19}}{104127350297911241532840}+\frac{\left(x^{36} y v_1^2+27 x^{35} y^2 v_1^2+366 x^{34} y^3 v_1^2+3315 x^{33} y^4 v_1^2+22491 x^{32} y^5 v_1^2+121380 x^{31} y^6 v_1^2+540192 x^{30} y^7 v_1^2+2029698 x^{29} y^8 v_1^2+6545000 x^{28} y^9 v_1^2+18330862 x^{27} y^{10} v_1^2+44997912 x^{26} y^{11} v_1^2+97498128 x^{25} y^{12} v_1^2+187497828 x^{24} y^{13} v_1^2+321425460 x^{23} y^{14} v_1^2+492852576 x^{22} y^{15} v_1^2+677672343 x^{21} y^{16} v_1^2+837124668 x^{20} y^{17} v_1^2+930138521 x^{19} y^{18} v_1^2+930138521 x^{18} y^{19} v_1^2+837124668 x^{17} y^{20} v_1^2+677672343 x^{16} y^{21} v_1^2+492852576 x^{15} y^{22} 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v_1^3+20475 x^4 y^{51} v_1^3+1320 x^3 y^{52} v_1^3+54 x^2 y^{53} v_1^3+x y^{54} v_1^3\right) d^{55}}{1129001324578697586834645174834954139048475361635681120538544010304000}+\frac{\left(x^{72} y v_1^4+90 x^{71} y^2 v_1^4+3450 x^{70} y^3 v_1^4+80850 x^{69} y^4 v_1^4+1347066 x^{68} y^5 v_1^4+17315928 x^{67} y^6 v_1^4+180622620 x^{66} y^7 v_1^4+1581473025 x^{65} y^8 v_1^4+11905273655 x^{64} y^9 v_1^4+78436292792 x^{63} y^{10} v_1^4+458445071448 x^{62} y^{11} v_1^4+2402533480620 x^{61} y^{12} v_1^4+11385746621760 x^{60} y^{13} v_1^4+49133339244720 x^{59} y^{14} v_1^4+194180198819616 x^{58} y^{15} v_1^4+706209892869411 x^{57} y^{16} v_1^4+2373172843438800 x^{56} y^{17} v_1^4+7394378629027600 x^{55} y^{18} v_1^4+21426541509274521 x^{54} y^{19} v_1^4+57890833193400408 x^{53} y^{20} v_1^4+146170722029709960 x^{52} y^{21} v_1^4+345595330733430600 x^{51} y^{22} v_1^4+766464846551414280 x^{50} y^{23} v_1^4+1596994784315685180 x^{49} y^{24} v_1^4+3130349122979634528 x^{48} y^{25} 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$(x+y) d+\frac{\left(x^{22} y v_1+11 x^{21} y^2 v_1+77 x^{20} y^3 v_1+385 x^{19} y^4 v_1+1463 x^{18} y^5 v_1+4389 x^{17} y^6 v_1+10659 x^{16} y^7 v_1+21318 x^{15} y^8 v_1+35530 x^{14} y^9 v_1+49742 x^{13} y^{10} v_1+58786 x^{12} y^{11} v_1+58786 x^{11} y^{12} v_1+49742 x^{10} y^{13} v_1+35530 x^9 y^{14} v_1+21318 x^8 y^{15} v_1+10659 x^7 y^{16} v_1+4389 x^6 y^{17} v_1+1463 x^5 y^{18} v_1+385 x^4 y^{19} v_1+77 x^3 y^{20} v_1+11 x^2 y^{21} v_1+x y^{22} v_1\right) d^{23}}{907846434775996175406740561328}+\frac{\left(x^{44} y v_1^2+33 x^{43} y^2 v_1^2+550 x^{42} y^3 v_1^2+6160 x^{41} y^4 v_1^2+51975 x^{40} y^5 v_1^2+350889 x^{39} y^6 v_1^2+1965612 x^{38} y^7 v_1^2+9357975 x^{37} y^8 v_1^2+38507205 x^{36} y^9 v_1^2+138675680 x^{35} y^{10} v_1^2+441299586 x^{34} y^{11} v_1^2+1250407613 x^{33} y^{12} v_1^2+3174161375 x^{32} y^{13} v_1^2+7255261530 x^{31} y^{14} v_1^2+14994228480 x^{30} y^{15} v_1^2+28114189059 x^{29} y^{16} v_1^2+47959503372 x^{28} y^{17} v_1^2+74603673375 x^{27} y^{18} 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y^{135} v_1^3+8740 x^3 y^{136} v_1^3+138 x^2 y^{137} v_1^3+x y^{138} v_1^3\right) d^{139}}{561699962819551524334627176256400317941991135997449283178714528863427682354650107828030627174491695576642570109689019157276850859934075371559586263407203691001727860055746571888002596713333605194689095847081316427799908501374205952}+O[d]^{140}$

8 Dec 2018 | categories: Photographs, Mathematics

Clifford & Marx

Karl Marx at Highgate (2018)

Clifford & Marx at Highgate (2018)

William Kingdon Clifford at Highgate (2018)

The headstones of Clifford (†1879) and Marx (†1883) seem to be in conversation at Highgate Cemetery in London.


26 Jun 2018 | categories: Mathematics, Aviation

Cayley Lane

Carl McTague on Cayley Lane (2018)

There are two things called the Cayley Plane,

  1. The projective plane over the octonions, named after the math­e­mat­ician Sir Arthur Cayley (1821–1895), and

  2. George Cayley’s Sketch of His “Boy-Carrier” (1849)
    George Cayley’s Sketch of His “Boy-Carrier” (1849)
    The ground­breaking aero­plane designed by his distant cousin, the aviation pioneer Sir George Cayley (1774–1857).

I could write a lot about the first. In fact, I’ve written a paper about it,

The Cayley Plane and String Bordism, Geometry & Topology 18-4 (2014), 2045–2078.

But last week I travelled to Brompton [outside Scarborough, UK] to pay homage to the second.

I visited Sir George’s workshop at Brompton Hall, where he designed his plane. [Cayley Lane, pictured above, runs beside it.] And I spent the night at Sir George’s annex Wydale Hall, now a retreat centre run by the Diocese of York.

Wydale Hall (2018)Vale of Pickering (2018)

After breakfast a kind 83-year-old pastor walked me to Brompton Dale, where in 1853 Sir George’s terrified coachman John Appleby flew the plane [afterwards saying “Please, Sir George, I wish to give notice. I was hired to drive, and not to fly.”]

Carl McTague at Brompton Dale (2018)

Brompton Dale (2018)

There is, as far as I know, no connection with the Brompton folding bicycle – my primary mode of transport, named by its inventor Andrew Ritchie after the Brompton Oratory in London, where Alfred Hitchcock got hitched, and which Ritchie could see from his workshop window while building prototypes in 1976 – outside the fact that Sir George invented the wire wheel for his plane.

Speaking of London, a few days earlier I visited Lincoln’s Inn, where in the 1840’s Sir Arthur would meet to discuss invariant theory with his friend JJ Sylvester – not to be confused with Sylvester II.


27 Nov 2017 | categories: Mathematics

A Dynamical Proof That the Ratio of Successive Fibonacci Numbers Approaches the Golden Ratio

In his 1611 essay On the Six-Cornered Snowflake, Johannes Kepler observed that the ratio of successive Fibonacci numbers $F_n/F_{n-1}$ approaches the golden ratio $\tfrac12(1+\sqrt5)$. [Recall that $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\ge2$.]

Many proofs have since been found. Here is a quick, dynamical one I thought of which I have been unable to find in the literature.

Proof: Consider the function $f(x)=(1+x)/x$. The sequence of ratios of successive Fibonacci numbers is the orbit of $F_2/F_1=1$ under $f$, i.e. $$ f(F_n/F_{n-1})=F_{n+1}/F_n. $$ On the other hand, $f$ has precisely two fixed points, the roots of $x^2-x-1$, one positive (the golden ratio), the other negative. Since $f'(x)=-1/x^2$, $f$ is a decreasing function for $x>0$. And $f(2)=3/2$ while $f(3/2)=5/3$. So $f$ maps the interval $[3/2,2]$ into itself. In fact, $f$ is a contracting mapping on $[3/2,2]$ (with Lipschitz constant $4/9$) since $$ |f(x)-f(y)|=|y-x|/|xy|\le\tfrac49|y-x| $$ for $x,y\in[3/2,2]$. By the Banach fixed point theorem, then, all orbits in $[3/2,2]$ approach the golden ratio. Although $1$ is not in $[3/2,2]$, $f(1)=2$ is, so the orbit of $1$ approaches the golden ratio. ∎

This proof can be extended to show that the conclusion holds even if the first two terms $F_1,F_2$ of the Fibonacci sequence are altered, provided their ratio $F_2/F_1$ is not $\tfrac12(1-\sqrt5)$ (the negative, unstable fixed point of $f$).

In a sense this explains why the limit is the golden ratio: being a fixed point of $f$ is the defining property of the golden ratio.


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