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Associated Legendre polynomials $P_\ell^m$
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Polynomials
$\ell$
$m$ 
$P_\ell^m(x)$, with $y=(1-x^2)^{1/2}$
0
0:
1
1
0:
x
1
1:
-y
1
-1:
1/2*y
2
0:
3/2*x^2 - 1/2
2
1:
-3*x*y
2
-1:
1/2*x*y
2
2:
3*y^2
2
-2:
1/8*y^2
3
0:
5/2*x^3 - 3/2*x
3
1:
-15/2*x^2*y + 3/2*y
3
-1:
5/8*x^2*y - 1/8*y
3
2:
15*x*y^2
3
-2:
1/8*x*y^2
3
3:
-15*y^3
3
-3:
1/48*y^3
4
0:
35/8*x^4 - 15/4*x^2 + 3/8
4
1:
-35/2*x^3*y + 15/2*x*y
4
-1:
7/8*x^3*y - 3/8*x*y
4
2:
105/2*x^2*y^2 - 15/2*y^2
4
-2:
7/48*x^2*y^2 - 1/48*y^2
4
3:
-105*x*y^3
4
-3:
1/48*x*y^3
4
4:
105*y^4
4
-4:
1/384*y^4
5
0:
63/8*x^5 - 35/4*x^3 + 15/8*x
5
1:
-315/8*x^4*y + 105/4*x^2*y - 15/8*y
5
-1:
21/16*x^4*y - 7/8*x^2*y + 1/16*y
5
2:
315/2*x^3*y^2 - 105/2*x*y^2
5
-2:
3/16*x^3*y^2 - 1/16*x*y^2
5
3:
-945/2*x^2*y^3 + 105/2*y^3
5
-3:
3/128*x^2*y^3 - 1/384*y^3
5
4:
945*x*y^4
5
-4:
1/384*x*y^4
5
5:
-945*y^5
5
-5:
1/3840*y^5
6
0:
231/16*x^6 - 315/16*x^4 + 105/16*x^2 - 5/16
6
1:
-693/8*x^5*y + 315/4*x^3*y - 105/8*x*y
6
-1:
33/16*x^5*y - 15/8*x^3*y + 5/16*x*y
6
2:
3465/8*x^4*y^2 - 945/4*x^2*y^2 + 105/8*y^2
6
-2:
33/128*x^4*y^2 - 9/64*x^2*y^2 + 1/128*y^2
6
3:
-3465/2*x^3*y^3 + 945/2*x*y^3
6
-3:
11/384*x^3*y^3 - 1/128*x*y^3
6
4:
10395/2*x^2*y^4 - 945/2*y^4
6
-4:
11/3840*x^2*y^4 - 1/3840*y^4
6
5:
-10395*x*y^5
6
-5:
1/3840*x*y^5
6
6:
10395*y^6
6
-6:
1/46080*y^6
7
0:
429/16*x^7 - 693/16*x^5 + 315/16*x^3 - 35/16*x
7
1:
-3003/16*x^6*y + 3465/16*x^4*y - 945/16*x^2*y + 35/16*y
7
-1:
429/128*x^6*y - 495/128*x^4*y + 135/128*x^2*y - 5/128*y
7
2:
9009/8*x^5*y^2 - 3465/4*x^3*y^2 + 945/8*x*y^2
7
-2:
143/384*x^5*y^2 - 55/192*x^3*y^2 + 5/128*x*y^2
7
3:
-45045/8*x^4*y^3 + 10395/4*x^2*y^3 - 945/8*y^3
7
-3:
143/3840*x^4*y^3 - 11/640*x^2*y^3 + 1/1280*y^3
7
4:
45045/2*x^3*y^4 - 10395/2*x*y^4
7
-4:
13/3840*x^3*y^4 - 1/1280*x*y^4
7
5:
-135135/2*x^2*y^5 + 10395/2*y^5
7
-5:
13/46080*x^2*y^5 - 1/46080*y^5
7
6:
135135*x*y^6
7
-6:
1/46080*x*y^6
7
7:
-135135*y^7
7
-7:
1/645120*y^7
8
0:
6435/128*x^8 - 3003/32*x^6 + 3465/64*x^4 - 315/32*x^2 + 35/128
8
1:
-6435/16*x^7*y + 9009/16*x^5*y - 3465/16*x^3*y + 315/16*x*y
8
-1:
715/128*x^7*y - 1001/128*x^5*y + 385/128*x^3*y - 35/128*x*y
8
2:
45045/16*x^6*y^2 - 45045/16*x^4*y^2 + 10395/16*x^2*y^2 - 315/16*y^2
8
-2:
143/256*x^6*y^2 - 143/256*x^4*y^2 + 33/256*x^2*y^2 - 1/256*y^2
8
3:
-135135/8*x^5*y^3 + 45045/4*x^3*y^3 - 10395/8*x*y^3
8
-3:
13/256*x^5*y^3 - 13/384*x^3*y^3 + 1/256*x*y^3
8
4:
675675/8*x^4*y^4 - 135135/4*x^2*y^4 + 10395/8*y^4
8
-4:
13/3072*x^4*y^4 - 13/7680*x^2*y^4 + 1/15360*y^4
8
5:
-675675/2*x^3*y^5 + 135135/2*x*y^5
8
-5:
1/3072*x^3*y^5 - 1/15360*x*y^5
8
6:
2027025/2*x^2*y^6 - 135135/2*y^6
8
-6:
1/43008*x^2*y^6 - 1/645120*y^6
8
7:
-2027025*x*y^7
8
-7:
1/645120*x*y^7
8
8:
2027025*y^8
8
-8:
1/10321920*y^8
9
0:
12155/128*x^9 - 6435/32*x^7 + 9009/64*x^5 - 1155/32*x^3 + 315/128*x
9
1:
-109395/128*x^8*y + 45045/32*x^6*y - 45045/64*x^4*y + 3465/32*x^2*y - 315/128*y
9
-1:
2431/256*x^8*y - 1001/64*x^6*y + 1001/128*x^4*y - 77/64*x^2*y + 7/256*y
9
2:
109395/16*x^7*y^2 - 135135/16*x^5*y^2 + 45045/16*x^3*y^2 - 3465/16*x*y^2
9
-2:
221/256*x^7*y^2 - 273/256*x^5*y^2 + 91/256*x^3*y^2 - 7/256*x*y^2
9
3:
-765765/16*x^6*y^3 + 675675/16*x^4*y^3 - 135135/16*x^2*y^3 + 3465/16*y^3
9
-3:
221/3072*x^6*y^3 - 65/1024*x^4*y^3 + 13/1024*x^2*y^3 - 1/3072*y^3
9
4:
2297295/8*x^5*y^4 - 675675/4*x^3*y^4 + 135135/8*x*y^4
9
-4:
17/3072*x^5*y^4 - 5/1536*x^3*y^4 + 1/3072*x*y^4
9
5:
-11486475/8*x^4*y^5 + 2027025/4*x^2*y^5 - 135135/8*y^5
9
-5:
17/43008*x^4*y^5 - 1/7168*x^2*y^5 + 1/215040*y^5
9
6:
11486475/2*x^3*y^6 - 2027025/2*x*y^6
9
-6:
17/645120*x^3*y^6 - 1/215040*x*y^6
9
7:
-34459425/2*x^2*y^7 + 2027025/2*y^7
9
-7:
17/10321920*x^2*y^7 - 1/10321920*y^7
9
8:
34459425*x*y^8
9
-8:
1/10321920*x*y^8
9
9:
-34459425*y^9
9
-9:
1/185794560*y^9
10
0:
46189/256*x^10 - 109395/256*x^8 + 45045/128*x^6 - 15015/128*x^4 + 3465/256*x^2 - 63/256
10
1:
-230945/128*x^9*y + 109395/32*x^7*y - 135135/64*x^5*y + 15015/32*x^3*y - 3465/128*x*y
10
-1:
4199/256*x^9*y - 1989/64*x^7*y + 2457/128*x^5*y - 273/64*x^3*y + 63/256*x*y
10
2:
2078505/128*x^8*y^2 - 765765/32*x^6*y^2 + 675675/64*x^4*y^2 - 45045/32*x^2*y^2 + 3465/128*y^2
10
-2:
4199/3072*x^8*y^2 - 1547/768*x^6*y^2 + 455/512*x^4*y^2 - 91/768*x^2*y^2 + 7/3072*y^2
10
3:
-2078505/16*x^7*y^3 + 2297295/16*x^5*y^3 - 675675/16*x^3*y^3 + 45045/16*x*y^3
10
-3:
323/3072*x^7*y^3 - 119/1024*x^5*y^3 + 35/1024*x^3*y^3 - 7/3072*x*y^3
10
4:
14549535/16*x^6*y^4 - 11486475/16*x^4*y^4 + 2027025/16*x^2*y^4 - 45045/16*y^4
10
-4:
323/43008*x^6*y^4 - 85/14336*x^4*y^4 + 15/14336*x^2*y^4 - 1/43008*y^4
10
5:
-43648605/8*x^5*y^5 + 11486475/4*x^3*y^5 - 2027025/8*x*y^5
10
-5:
323/645120*x^5*y^5 - 17/64512*x^3*y^5 + 1/43008*x*y^5
10
6:
218243025/8*x^4*y^6 - 34459425/4*x^2*y^6 + 2027025/8*y^6
10
-6:
323/10321920*x^4*y^6 - 17/1720320*x^2*y^6 + 1/3440640*y^6
10
7:
-218243025/2*x^3*y^7 + 34459425/2*x*y^7
10
-7:
19/10321920*x^3*y^7 - 1/3440640*x*y^7
10
8:
654729075/2*x^2*y^8 - 34459425/2*y^8
10
-8:
19/185794560*x^2*y^8 - 1/185794560*y^8
10
9:
-654729075*x*y^9
10
-9:
1/185794560*x*y^9
10
10:
654729075*y^10
10
-10:
1/3715891200*y^10
11
0:
88179/256*x^11 - 230945/256*x^9 + 109395/128*x^7 - 45045/128*x^5 + 15015/256*x^3 - 693/256*x
11
1:
-969969/256*x^10*y + 2078505/256*x^8*y - 765765/128*x^6*y + 225225/128*x^4*y - 45045/256*x^2*y + 693/256*y
11
-1:
29393/1024*x^10*y - 62985/1024*x^8*y + 23205/512*x^6*y - 6825/512*x^4*y + 1365/1024*x^2*y - 21/1024*y
11
2:
4849845/128*x^9*y^2 - 2078505/32*x^7*y^2 + 2297295/64*x^5*y^2 - 225225/32*x^3*y^2 + 45045/128*x*y^2
11
-2:
2261/1024*x^9*y^2 - 969/256*x^7*y^2 + 1071/512*x^5*y^2 - 105/256*x^3*y^2 + 21/1024*x*y^2
11
3:
-43648605/128*x^8*y^3 + 14549535/32*x^6*y^3 - 11486475/64*x^4*y^3 + 675675/32*x^2*y^3 - 45045/128*y^3
11
-3:
323/2048*x^8*y^3 - 323/1536*x^6*y^3 + 85/1024*x^4*y^3 - 5/512*x^2*y^3 + 1/6144*y^3
11
4:
43648605/16*x^7*y^4 - 43648605/16*x^5*y^4 + 11486475/16*x^3*y^4 - 675675/16*x*y^4
11
-4:
323/30720*x^7*y^4 - 323/30720*x^5*y^4 + 17/6144*x^3*y^4 - 1/6144*x*y^4
11
5:
-305540235/16*x^6*y^5 + 218243025/16*x^4*y^5 - 34459425/16*x^2*y^5 + 675675/16*y^5
11
-5:
323/491520*x^6*y^5 - 323/688128*x^4*y^5 + 17/229376*x^2*y^5 - 1/688128*y^5
11
6:
916620705/8*x^5*y^6 - 218243025/4*x^3*y^6 + 34459425/8*x*y^6
11
-6:
19/491520*x^5*y^6 - 19/1032192*x^3*y^6 + 1/688128*x*y^6
11
7:
-4583103525/8*x^4*y^7 + 654729075/4*x^2*y^7 - 34459425/8*y^7
11
-7:
19/8847360*x^4*y^7 - 19/30965760*x^2*y^7 + 1/61931520*y^7
11
8:
4583103525/2*x^3*y^8 - 654729075/2*x*y^8
11
-8:
1/8847360*x^3*y^8 - 1/61931520*x*y^8
11
9:
-13749310575/2*x^2*y^9 + 654729075/2*y^9
11
-9:
1/176947200*x^2*y^9 - 1/3715891200*y^9
11
10:
13749310575*x*y^10
11
-10:
1/3715891200*x*y^10
11
11:
-13749310575*y^11
11
-11:
1/81749606400*y^11
12
0:
676039/1024*x^12 - 969969/512*x^10 + 2078505/1024*x^8 - 255255/256*x^6 + 225225/1024*x^4 - 9009/512*x^2 + 231/1024
12
1:
-2028117/256*x^11*y + 4849845/256*x^9*y - 2078505/128*x^7*y + 765765/128*x^5*y - 225225/256*x^3*y + 9009/256*x*y
12
-1:
52003/1024*x^11*y - 124355/1024*x^9*y + 53295/512*x^7*y - 19635/512*x^5*y + 5775/1024*x^3*y - 231/1024*x*y
12
2:
22309287/256*x^10*y^2 - 43648605/256*x^8*y^2 + 14549535/128*x^6*y^2 - 3828825/128*x^4*y^2 + 675675/256*x^2*y^2 - 9009/256*y^2
12
-2:
7429/2048*x^10*y^2 - 14535/2048*x^8*y^2 + 4845/1024*x^6*y^2 - 1275/1024*x^4*y^2 + 225/2048*x^2*y^2 - 3/2048*y^2
12
3:
-111546435/128*x^9*y^3 + 43648605/32*x^7*y^3 - 43648605/64*x^5*y^3 + 3828825/32*x^3*y^3 - 675675/128*x*y^3
12
-3:
7429/30720*x^9*y^3 - 969/2560*x^7*y^3 + 969/5120*x^5*y^3 - 17/512*x^3*y^3 + 3/2048*x*y^3
12
4:
1003917915/128*x^8*y^4 - 305540235/32*x^6*y^4 + 218243025/64*x^4*y^4 - 11486475/32*x^2*y^4 + 675675/128*y^4
12
-4:
7429/491520*x^8*y^4 - 2261/122880*x^6*y^4 + 323/49152*x^4*y^4 - 17/24576*x^2*y^4 + 1/98304*y^4
12
5:
-1003917915/16*x^7*y^5 + 916620705/16*x^5*y^5 - 218243025/16*x^3*y^5 + 11486475/16*x*y^5
12
-5:
437/491520*x^7*y^5 - 133/163840*x^5*y^5 + 19/98304*x^3*y^5 - 1/98304*x*y^5
12
6:
7027425405/16*x^6*y^6 - 4583103525/16*x^4*y^6 + 654729075/16*x^2*y^6 - 11486475/16*y^6
12
-6:
437/8847360*x^6*y^6 - 19/589824*x^4*y^6 + 19/4128768*x^2*y^6 - 1/12386304*y^6
12
7:
-21082276215/8*x^5*y^7 + 4583103525/4*x^3*y^7 - 654729075/8*x*y^7
12
-7:
23/8847360*x^5*y^7 - 1/884736*x^3*y^7 + 1/12386304*x*y^7
12
8:
105411381075/8*x^4*y^8 - 13749310575/4*x^2*y^8 + 654729075/8*y^8
12
-8:
23/176947200*x^4*y^8 - 1/29491200*x^2*y^8 + 1/1238630400*y^8
12
9:
-105411381075/2*x^3*y^9 + 13749310575/2*x*y^9
12
-9:
23/3715891200*x^3*y^9 - 1/1238630400*x*y^9
12
10:
316234143225/2*x^2*y^10 - 13749310575/2*y^10
12
-10:
23/81749606400*x^2*y^10 - 1/81749606400*y^10
12
11:
-316234143225*x*y^11
12
-11:
1/81749606400*x*y^11
12
12:
316234143225*y^12
12
-12:
1/1961990553600*y^12
13
0:
1300075/1024*x^13 - 2028117/512*x^11 + 4849845/1024*x^9 - 692835/256*x^7 + 765765/1024*x^5 - 45045/512*x^3 + 3003/1024*x
13
1:
-16900975/1024*x^12*y + 22309287/512*x^10*y - 43648605/1024*x^8*y + 4849845/256*x^6*y - 3828825/1024*x^4*y + 135135/512*x^2*y - 3003/1024*y
13
-1:
185725/2048*x^12*y - 245157/1024*x^10*y + 479655/2048*x^8*y - 53295/512*x^6*y + 42075/2048*x^4*y - 1485/1024*x^2*y + 33/2048*y
13
2:
50702925/256*x^11*y^2 - 111546435/256*x^9*y^2 + 43648605/128*x^7*y^2 - 14549535/128*x^5*y^2 + 3828825/256*x^3*y^2 - 135135/256*x*y^2
13
-2:
37145/6144*x^11*y^2 - 81719/6144*x^9*y^2 + 10659/1024*x^7*y^2 - 3553/1024*x^5*y^2 + 935/2048*x^3*y^2 - 33/2048*x*y^2
13
3:
-557732175/256*x^10*y^3 + 1003917915/256*x^8*y^3 - 305540235/128*x^6*y^3 + 72747675/128*x^4*y^3 - 11486475/256*x^2*y^3 + 135135/256*y^3
13
-3:
37145/98304*x^10*y^3 - 22287/32768*x^8*y^3 + 6783/16384*x^6*y^3 - 1615/16384*x^4*y^3 + 255/32768*x^2*y^3 - 3/32768*y^3
13
4:
2788660875/128*x^9*y^4 - 1003917915/32*x^7*y^4 + 916620705/64*x^5*y^4 - 72747675/32*x^3*y^4 + 11486475/128*x*y^4
13
-4:
2185/98304*x^9*y^4 - 1311/40960*x^7*y^4 + 1197/81920*x^5*y^4 - 19/8192*x^3*y^4 + 3/32768*x*y^4
13
5:
-25097947875/128*x^8*y^5 + 7027425405/32*x^6*y^5 - 4583103525/64*x^4*y^5 + 218243025/32*x^2*y^5 - 11486475/128*y^5
13
-5:
2185/1769472*x^8*y^5 - 3059/2211840*x^6*y^5 + 133/294912*x^4*y^5 - 19/442368*x^2*y^5 + 1/1769472*y^5
13
6:
25097947875/16*x^7*y^6 - 21082276215/16*x^5*y^6 + 4583103525/16*x^3*y^6 - 218243025/16*x*y^6
13
-6:
115/1769472*x^7*y^6 - 161/2949120*x^5*y^6 + 7/589824*x^3*y^6 - 1/1769472*x*y^6
13
7:
-175685635125/16*x^6*y^7 + 105411381075/16*x^4*y^7 - 13749310575/16*x^2*y^7 + 218243025/16*y^7
13
-7:
23/7077888*x^6*y^7 - 23/11796480*x^4*y^7 + 1/3932160*x^2*y^7 - 1/247726080*y^7
13
8:
527056905375/8*x^5*y^8 - 105411381075/4*x^3*y^8 + 13749310575/8*x*y^8
13
-8:
23/148635648*x^5*y^8 - 23/371589120*x^3*y^8 + 1/247726080*x*y^8
13
9:
-2635284526875/8*x^4*y^9 + 316234143225/4*x^2*y^9 - 13749310575/8*y^9
13
-9:
23/3269984256*x^4*y^9 - 23/13624934400*x^2*y^9 + 1/27249868800*y^9
13
10:
2635284526875/2*x^3*y^10 - 316234143225/2*x*y^10
13
-10:
1/3269984256*x^3*y^10 - 1/27249868800*x*y^10
13
11:
-7905853580625/2*x^2*y^11 + 316234143225/2*y^11
13
-11:
1/78479622144*x^2*y^11 - 1/1961990553600*y^11
13
12:
7905853580625*x*y^12
13
-12:
1/1961990553600*x*y^12
13
13:
-7905853580625*y^13
13
-13:
1/51011754393600*y^13
14
0:
5014575/2048*x^14 - 16900975/2048*x^12 + 22309287/2048*x^10 - 14549535/2048*x^8 + 4849845/2048*x^6 - 765765/2048*x^4 + 45045/2048*x^2 - 429/2048
14
1:
-35102025/1024*x^13*y + 50702925/512*x^11*y - 111546435/1024*x^9*y + 14549535/256*x^7*y - 14549535/1024*x^5*y + 765765/512*x^3*y - 45045/1024*x*y
14
-1:
334305/2048*x^13*y - 482885/1024*x^11*y + 1062347/2048*x^9*y - 138567/512*x^7*y + 138567/2048*x^5*y - 7293/1024*x^3*y + 429/2048*x*y
14
2:
456326325/1024*x^12*y^2 - 557732175/512*x^10*y^2 + 1003917915/1024*x^8*y^2 - 101846745/256*x^6*y^2 + 72747675/1024*x^4*y^2 - 2297295/512*x^2*y^2 + 45045/1024*y^2
14
-2:
334305/32768*x^12*y^2 - 408595/16384*x^10*y^2 + 735471/32768*x^8*y^2 - 74613/8192*x^6*y^2 + 53295/32768*x^4*y^2 - 1683/16384*x^2*y^2 + 33/32768*y^2
14
3:
-1368978975/256*x^11*y^3 + 2788660875/256*x^9*y^3 - 1003917915/128*x^7*y^3 + 305540235/128*x^5*y^3 - 72747675/256*x^3*y^3 + 2297295/256*x*y^3
14
-3:
19665/32768*x^11*y^3 - 120175/98304*x^9*y^3 + 14421/16384*x^7*y^3 - 4389/16384*x^5*y^3 + 1045/32768*x^3*y^3 - 33/32768*x*y^3
14
4:
15058768725/256*x^10*y^4 - 25097947875/256*x^8*y^4 + 7027425405/128*x^6*y^4 - 1527701175/128*x^4*y^4 + 218243025/256*x^2*y^4 - 2297295/256*y^4
14
-4:
2185/65536*x^10*y^4 - 10925/196608*x^8*y^4 + 3059/98304*x^6*y^4 - 665/98304*x^4*y^4 + 95/196608*x^2*y^4 - 1/196608*y^4
14
5:
-75293843625/128*x^9*y^5 + 25097947875/32*x^7*y^5 - 21082276215/64*x^5*y^5 + 1527701175/32*x^3*y^5 - 218243025/128*x*y^5
14
-5:
115/65536*x^9*y^5 - 115/49152*x^7*y^5 + 161/163840*x^5*y^5 - 7/49152*x^3*y^5 + 1/196608*x*y^5
14
6:
677644592625/128*x^8*y^6 - 175685635125/32*x^6*y^6 + 105411381075/64*x^4*y^6 - 4583103525/32*x^2*y^6 + 218243025/128*y^6
14
-6:
23/262144*x^8*y^6 - 161/1769472*x^6*y^6 + 161/5898240*x^4*y^6 - 7/2949120*x^2*y^6 + 1/35389440*y^6
14
7:
-677644592625/16*x^7*y^7 + 527056905375/16*x^5*y^7 - 105411381075/16*x^3*y^7 + 4583103525/16*x*y^7
14
-7:
23/5505024*x^7*y^7 - 23/7077888*x^5*y^7 + 23/35389440*x^3*y^7 - 1/35389440*x*y^7
14
8:
4743512148375/16*x^6*y^8 - 2635284526875/16*x^4*y^8 + 316234143225/16*x^2*y^8 - 4583103525/16*y^8
14
-8:
23/121110528*x^6*y^8 - 115/1089994752*x^4*y^8 + 23/1816657920*x^2*y^8 - 1/5449973760*y^8
14
9:
-14230536445125/8*x^5*y^9 + 2635284526875/4*x^3*y^9 - 316234143225/8*x*y^9
14
-9:
1/121110528*x^5*y^9 - 5/1634992128*x^3*y^9 + 1/5449973760*x*y^9
14
10:
71152682225625/8*x^4*y^10 - 7905853580625/4*x^2*y^10 + 316234143225/8*y^10
14
-10:
1/2906652672*x^4*y^10 - 1/13079937024*x^2*y^10 + 1/653996851200*y^10
14
11:
-71152682225625/2*x^3*y^11 + 7905853580625/2*x*y^11
14
-11:
1/72666316800*x^3*y^11 - 1/653996851200*x*y^11
14
12:
213458046676875/2*x^2*y^12 - 7905853580625/2*y^12
14
-12:
1/1889324236800*x^2*y^12 - 1/51011754393600*y^12
14
13:
-213458046676875*x*y^13
14
-13:
1/51011754393600*x*y^13
14
14:
213458046676875*y^14
14
-14:
1/1428329123020800*y^14
15
0:
9694845/2048*x^15 - 35102025/2048*x^13 + 50702925/2048*x^11 - 37182145/2048*x^9 + 14549535/2048*x^7 - 2909907/2048*x^5 + 255255/2048*x^3 - 6435/2048*x
15
1:
-145422675/2048*x^14*y + 456326325/2048*x^12*y - 557732175/2048*x^10*y + 334639305/2048*x^8*y - 101846745/2048*x^6*y + 14549535/2048*x^4*y - 765765/2048*x^2*y + 6435/2048*y
15
-1:
9694845/32768*x^14*y - 30421755/32768*x^12*y + 37182145/32768*x^10*y - 22309287/32768*x^8*y + 6789783/32768*x^6*y - 969969/32768*x^4*y + 51051/32768*x^2*y - 429/32768*y
15
2:
1017958725/1024*x^13*y^2 - 1368978975/512*x^11*y^2 + 2788660875/1024*x^9*y^2 - 334639305/256*x^7*y^2 + 305540235/1024*x^5*y^2 - 14549535/512*x^3*y^2 + 765765/1024*x*y^2
15
-2:
570285/32768*x^13*y^2 - 766935/16384*x^11*y^2 + 1562275/32768*x^9*y^2 - 187473/8192*x^7*y^2 + 171171/32768*x^5*y^2 - 8151/16384*x^3*y^2 + 429/32768*x*y^2
15
3:
-13233463425/1024*x^12*y^3 + 15058768725/512*x^10*y^3 - 25097947875/1024*x^8*y^3 + 2342475135/256*x^6*y^3 - 1527701175/1024*x^4*y^3 + 43648605/512*x^2*y^3 - 765765/1024*y^3
15
-3:
63365/65536*x^12*y^3 - 72105/32768*x^10*y^3 + 120175/65536*x^8*y^3 - 33649/49152*x^6*y^3 + 7315/65536*x^4*y^3 - 209/32768*x^2*y^3 + 11/196608*y^3
15
4:
39700390275/256*x^11*y^4 - 75293843625/256*x^9*y^4 + 25097947875/128*x^7*y^4 - 7027425405/128*x^5*y^4 + 1527701175/256*x^3*y^4 - 43648605/256*x*y^4
15
-4:
3335/65536*x^11*y^4 - 6325/65536*x^9*y^4 + 6325/98304*x^7*y^4 - 1771/98304*x^5*y^4 + 385/196608*x^3*y^4 - 11/196608*x*y^4
15
5:
-436704293025/256*x^10*y^5 + 677644592625/256*x^8*y^5 - 175685635125/128*x^6*y^5 + 35137127025/128*x^4*y^5 - 4583103525/256*x^2*y^5 + 43648605/256*y^5
15
-5:
667/262144*x^10*y^5 - 1035/262144*x^8*y^5 + 805/393216*x^6*y^5 - 161/393216*x^4*y^5 + 7/262144*x^2*y^5 - 1/3932160*y^5
15
6:
2183521465125/128*x^9*y^6 - 677644592625/32*x^7*y^6 + 527056905375/64*x^5*y^6 - 35137127025/32*x^3*y^6 + 4583103525/128*x*y^6
15
-6:
667/5505024*x^9*y^6 - 69/458752*x^7*y^6 + 23/393216*x^5*y^6 - 23/2949120*x^3*y^6 + 1/3932160*x*y^6
15
7:
-19651693186125/128*x^8*y^7 + 4743512148375/32*x^6*y^7 - 2635284526875/64*x^4*y^7 + 105411381075/32*x^2*y^7 - 4583103525/128*y^7
15
-7:
667/121110528*x^8*y^7 - 23/4325376*x^6*y^7 + 115/77856768*x^4*y^7 - 23/194641920*x^2*y^7 + 1/778567680*y^7
15
8:
19651693186125/16*x^7*y^8 - 14230536445125/16*x^5*y^8 + 2635284526875/16*x^3*y^8 - 105411381075/16*x*y^8
15
-8:
29/121110528*x^7*y^8 - 1/5767168*x^5*y^8 + 5/155713536*x^3*y^8 - 1/778567680*x*y^8
15
9:
-137561852302875/16*x^6*y^9 + 71152682225625/16*x^4*y^9 - 7905853580625/16*x^2*y^9 + 105411381075/16*y^9
15
-9:
29/2906652672*x^6*y^9 - 5/968884224*x^4*y^9 + 5/8719958016*x^2*y^9 - 1/130799370240*y^9
15
10:
412685556908625/8*x^5*y^10 - 71152682225625/4*x^3*y^10 + 7905853580625/8*x*y^10
15
-10:
29/72666316800*x^5*y^10 - 1/7266631680*x^3*y^10 + 1/130799370240*x*y^10
15
11:
-2063427784543125/8*x^4*y^11 + 213458046676875/4*x^2*y^11 - 7905853580625/8*y^11
15
-11:
29/1889324236800*x^4*y^11 - 1/314887372800*x^2*y^11 + 1/17003918131200*y^11
15
12:
2063427784543125/2*x^3*y^12 - 213458046676875/2*x*y^12
15
-12:
29/51011754393600*x^3*y^12 - 1/17003918131200*x*y^12
15
13:
-6190283353629375/2*x^2*y^13 + 213458046676875/2*y^13
15
-13:
29/1428329123020800*x^2*y^13 - 1/1428329123020800*y^13
15
14:
6190283353629375*x*y^14
15
-14:
1/1428329123020800*x*y^14
15
15:
-6190283353629375*y^15
15
-15:
1/42849873690624000*y^15
16
0:
300540195/32768*x^16 - 145422675/4096*x^14 + 456326325/8192*x^12 - 185910725/4096*x^10 + 334639305/16384*x^8 - 20369349/4096*x^6 + 4849845/8192*x^4 - 109395/4096*x^2 + 6435/32768
16
1:
-300540195/2048*x^15*y + 1017958725/2048*x^13*y - 1368978975/2048*x^11*y + 929553625/2048*x^9*y - 334639305/2048*x^7*y + 61108047/2048*x^5*y - 4849845/2048*x^3*y + 109395/2048*x*y
16
-1:
17678835/32768*x^15*y - 59879925/32768*x^13*y + 80528175/32768*x^11*y - 54679625/32768*x^9*y + 19684665/32768*x^7*y - 3594591/32768*x^5*y + 285285/32768*x^3*y - 6435/32768*x*y
16
2:
4508102925/2048*x^14*y^2 - 13233463425/2048*x^12*y^2 + 15058768725/2048*x^10*y^2 - 8365982625/2048*x^8*y^2 + 2342475135/2048*x^6*y^2 - 305540235/2048*x^4*y^2 + 14549535/2048*x^2*y^2 - 109395/2048*y^2
16
-2:
1964315/65536*x^14*y^2 - 5766215/65536*x^12*y^2 + 6561555/65536*x^10*y^2 - 10935925/196608*x^8*y^2 + 3062059/196608*x^6*y^2 - 133133/65536*x^4*y^2 + 19019/196608*x^2*y^2 - 143/196608*y^2
16
3:
-31556720475/1024*x^13*y^3 + 39700390275/512*x^11*y^3 - 75293843625/1024*x^9*y^3 + 8365982625/256*x^7*y^3 - 7027425405/1024*x^5*y^3 + 305540235/512*x^3*y^3 - 14549535/1024*x*y^3
16
-3:
103385/65536*x^13*y^3 - 130065/32768*x^11*y^3 + 246675/65536*x^9*y^3 - 82225/49152*x^7*y^3 + 23023/65536*x^5*y^3 - 1001/32768*x^3*y^3 + 143/196608*x*y^3
16
4:
410237366175/1024*x^12*y^4 - 436704293025/512*x^10*y^4 + 677644592625/1024*x^8*y^4 - 58561878375/256*x^6*y^4 + 35137127025/1024*x^4*y^4 - 916620705/512*x^2*y^4 + 14549535/1024*y^4
16
-4:
20677/262144*x^12*y^4 - 22011/131072*x^10*y^4 + 34155/262144*x^8*y^4 - 8855/196608*x^6*y^4 + 1771/262144*x^4*y^4 - 231/655360*x^2*y^4 + 11/3932160*y^4
16
5:
-1230712098525/256*x^11*y^5 + 2183521465125/256*x^9*y^5 - 677644592625/128*x^7*y^5 + 175685635125/128*x^5*y^5 - 35137127025/256*x^3*y^5 + 916620705/256*x*y^5
16
-5:
20677/5505024*x^11*y^5 - 36685/5505024*x^9*y^5 + 3795/917504*x^7*y^5 - 1265/1179648*x^5*y^5 + 253/2359296*x^3*y^5 - 11/3932160*x*y^5
16
6:
13537833083775/256*x^10*y^6 - 19651693186125/256*x^8*y^6 + 4743512148375/128*x^6*y^6 - 878428175625/128*x^4*y^6 + 105411381075/256*x^2*y^6 - 916620705/256*y^6
16
-6:
20677/121110528*x^10*y^6 - 10005/40370176*x^8*y^6 + 345/2883584*x^6*y^6 - 575/25952256*x^4*y^6 + 23/17301504*x^2*y^6 - 1/86507520*y^6
16
7:
-67689165418875/128*x^9*y^7 + 19651693186125/32*x^7*y^7 - 14230536445125/64*x^5*y^7 + 878428175625/32*x^3*y^7 - 105411381075/128*x*y^7
16
-7:
899/121110528*x^9*y^7 - 87/10092544*x^7*y^7 + 9/2883584*x^5*y^7 - 5/12976128*x^3*y^7 + 1/86507520*x*y^7
16
8:
609202488769875/128*x^8*y^8 - 137561852302875/32*x^6*y^8 + 71152682225625/64*x^4*y^8 - 2635284526875/32*x^2*y^8 + 105411381075/128*y^8
16
-8:
899/2906652672*x^8*y^8 - 29/103809024*x^6*y^8 + 5/69206016*x^4*y^8 - 5/934281216*x^2*y^8 + 1/18685624320*y^8
16
9:
-609202488769875/16*x^7*y^9 + 412685556908625/16*x^5*y^9 - 71152682225625/16*x^3*y^9 + 2635284526875/16*x*y^9
16
-9:
899/72666316800*x^7*y^9 - 29/3460300800*x^5*y^9 + 1/692060160*x^3*y^9 - 1/18685624320*x*y^9
16
10:
4264417421389125/16*x^6*y^10 - 2063427784543125/16*x^4*y^10 + 213458046676875/16*x^2*y^10 - 2635284526875/16*y^10
16
-10:
899/1889324236800*x^6*y^10 - 29/125954949120*x^4*y^10 + 1/41984983040*x^2*y^10 - 1/3400783626240*y^10
16
11:
-12793252264167375/8*x^5*y^11 + 2063427784543125/4*x^3*y^11 - 213458046676875/8*x*y^11
16
-11:
899/51011754393600*x^5*y^11 - 29/5101175439360*x^3*y^11 + 1/3400783626240*x*y^11
16
12:
63966261320836875/8*x^4*y^12 - 6190283353629375/4*x^2*y^12 + 213458046676875/8*y^12
16
-12:
899/1428329123020800*x^4*y^12 - 29/238054853836800*x^2*y^12 + 1/476109707673600*y^12
16
13:
-63966261320836875/2*x^3*y^13 + 6190283353629375/2*x*y^13
16
-13:
31/1428329123020800*x^3*y^13 - 1/476109707673600*x*y^13
16
14:
191898783962510625/2*x^2*y^14 - 6190283353629375/2*y^14
16
-14:
31/42849873690624000*x^2*y^14 - 1/42849873690624000*y^14
16
15:
-191898783962510625*x*y^15
16
-15:
1/42849873690624000*x*y^15
16
16:
191898783962510625*y^16
16
-16:
1/1371195958099968000*y^16
17
0:
583401555/32768*x^17 - 300540195/4096*x^15 + 1017958725/8192*x^13 - 456326325/4096*x^11 + 929553625/16384*x^9 - 66927861/4096*x^7 + 20369349/8192*x^5 - 692835/4096*x^3 + 109395/32768*x
17
1:
-9917826435/32768*x^16*y + 4508102925/4096*x^14*y - 13233463425/8192*x^12*y + 5019589575/4096*x^10*y - 8365982625/16384*x^8*y + 468495027/4096*x^6*y - 101846745/8192*x^4*y + 2078505/4096*x^2*y - 109395/32768*y
17
-1:
64822395/65536*x^16*y - 29464725/8192*x^14*y + 86493225/16384*x^12*y - 32807775/8192*x^10*y + 54679625/32768*x^8*y - 3062059/8192*x^6*y + 665665/16384*x^4*y - 13585/8192*x^2*y + 715/65536*y
17
2:
9917826435/2048*x^15*y^2 - 31556720475/2048*x^13*y^2 + 39700390275/2048*x^11*y^2 - 25097947875/2048*x^9*y^2 + 8365982625/2048*x^7*y^2 - 1405485081/2048*x^5*y^2 + 101846745/2048*x^3*y^2 - 2078505/2048*x*y^2
17
-2:
3411705/65536*x^15*y^2 - 10855425/65536*x^13*y^2 + 13656825/65536*x^11*y^2 - 8633625/65536*x^9*y^2 + 2877875/65536*x^7*y^2 - 483483/65536*x^5*y^2 + 35035/65536*x^3*y^2 - 715/65536*x*y^2
17
3:
-148767396525/2048*x^14*y^3 + 410237366175/2048*x^12*y^3 - 436704293025/2048*x^10*y^3 + 225881530875/2048*x^8*y^3 - 58561878375/2048*x^6*y^3 + 7027425405/2048*x^4*y^3 - 305540235/2048*x^2*y^3 + 2078505/2048*y^3
17
-3:
682341/262144*x^14*y^3 - 1881607/262144*x^12*y^3 + 2003001/262144*x^10*y^3 - 1036035/262144*x^8*y^3 + 805805/786432*x^6*y^3 - 161161/1310720*x^4*y^3 + 7007/1310720*x^2*y^3 - 143/3932160*y^3
17
4:
1041371775675/1024*x^13*y^4 - 1230712098525/512*x^11*y^4 + 2183521465125/1024*x^9*y^4 - 225881530875/256*x^7*y^4 + 175685635125/1024*x^5*y^4 - 7027425405/512*x^3*y^4 + 305540235/1024*x*y^4
17
-4:
227447/1835008*x^13*y^4 - 268801/917504*x^11*y^4 + 476905/1835008*x^9*y^4 - 49335/458752*x^7*y^4 + 16445/786432*x^5*y^4 - 3289/1966080*x^3*y^4 + 143/3932160*x*y^4
17
5:
-13537833083775/1024*x^12*y^5 + 13537833083775/512*x^10*y^5 - 19651693186125/1024*x^8*y^5 + 1581170716125/256*x^6*y^5 - 878428175625/1024*x^4*y^5 + 21082276215/512*x^2*y^5 - 305540235/1024*y^5
17
-5:
20677/3670016*x^12*y^5 - 20677/1835008*x^10*y^5 + 30015/3670016*x^8*y^5 - 345/131072*x^6*y^5 + 575/1572864*x^4*y^5 - 23/1310720*x^2*y^5 + 1/7864320*y^5
17
6:
40613499251325/256*x^11*y^6 - 67689165418875/256*x^9*y^6 + 19651693186125/128*x^7*y^6 - 4743512148375/128*x^5*y^6 + 878428175625/256*x^3*y^6 - 21082276215/256*x*y^6
17
-6:
899/3670016*x^11*y^6 - 4495/11010048*x^9*y^6 + 435/1835008*x^7*y^6 - 15/262144*x^5*y^6 + 25/4718592*x^3*y^6 - 1/7864320*x*y^6
17
7:
-446748491764575/256*x^10*y^7 + 609202488769875/256*x^8*y^7 - 137561852302875/128*x^6*y^7 + 23717560741875/128*x^4*y^7 - 2635284526875/256*x^2*y^7 + 21082276215/256*y^7
17
-7:
899/88080384*x^10*y^7 - 4495/322961408*x^8*y^7 + 145/23068672*x^6*y^7 - 25/23068672*x^4*y^7 + 25/415236096*x^2*y^7 - 1/2076180480*y^7
17
8:
2233742458822875/128*x^9*y^8 - 609202488769875/32*x^7*y^8 + 412685556908625/64*x^5*y^8 - 23717560741875/32*x^3*y^8 + 2635284526875/128*x*y^8
17
-8:
899/2202009600*x^9*y^8 - 899/2018508800*x^7*y^8 + 87/576716800*x^5*y^8 - 1/57671680*x^3*y^8 + 1/2076180480*x*y^8
17
9:
-20103682129405875/128*x^8*y^9 + 4264417421389125/32*x^6*y^9 - 2063427784543125/64*x^4*y^9 + 71152682225625/32*x^2*y^9 - 2635284526875/128*y^9
17
-9:
899/57252249600*x^8*y^9 - 899/67475865600*x^6*y^9 + 29/8996782080*x^4*y^9 - 1/4498391040*x^2*y^9 + 1/485826232320*y^9
17
10:
20103682129405875/16*x^7*y^10 - 12793252264167375/16*x^5*y^10 + 2063427784543125/16*x^3*y^10 - 71152682225625/16*x*y^10
17
-10:
899/1545810739200*x^7*y^10 - 899/2429131161600*x^5*y^10 + 29/485826232320*x^3*y^10 - 1/485826232320*x*y^10
17
11:
-140725774905841125/16*x^6*y^11 + 63966261320836875/16*x^4*y^11 - 6190283353629375/16*x^2*y^11 + 71152682225625/16*y^11
17
-11:
899/43282700697600*x^6*y^11 - 899/95221941534720*x^4*y^11 + 29/31740647178240*x^2*y^11 - 1/95221941534720*y^11
17
12:
422177324717523375/8*x^5*y^12 - 63966261320836875/4*x^3*y^12 + 6190283353629375/8*x*y^12
17
-12:
31/43282700697600*x^5*y^12 - 31/142832912302080*x^3*y^12 + 1/95221941534720*x*y^12
17
13:
-2110886623587616875/8*x^4*y^13 + 191898783962510625/4*x^2*y^13 - 6190283353629375/8*y^13
17
-13:
31/1298481020928000*x^4*y^13 - 31/7141645615104000*x^2*y^13 + 1/14283291230208000*y^13
17
14:
2110886623587616875/2*x^3*y^14 - 191898783962510625/2*x*y^14
17
-14:
1/1298481020928000*x^3*y^14 - 1/14283291230208000*x*y^14
17
15:
-6332659870762850625/2*x^2*y^15 + 191898783962510625/2*y^15
17
-15:
1/41551392669696000*x^2*y^15 - 1/1371195958099968000*y^15
17
16:
6332659870762850625*x*y^16
17
-16:
1/1371195958099968000*x*y^16
17
17:
-6332659870762850625*y^17
17
-17:
1/46620662575398912000*y^17
18
0:
2268783825/65536*x^18 - 9917826435/65536*x^16 + 4508102925/16384*x^14 - 4411154475/16384*x^12 + 5019589575/32768*x^10 - 1673196525/32768*x^8 + 156165009/16384*x^6 - 14549535/16384*x^4 + 2078505/65536*x^2 - 12155/65536
18
1:
-20419054425/32768*x^17*y + 9917826435/4096*x^15*y - 31556720475/8192*x^13*y + 13233463425/4096*x^11*y - 25097947875/16384*x^9*y + 1673196525/4096*x^7*y - 468495027/8192*x^5*y + 14549535/4096*x^3*y - 2078505/32768*x*y
18
-1:
119409675/65536*x^17*y - 57998985/8192*x^15*y + 184542225/16384*x^13*y - 77388675/8192*x^11*y + 146771625/32768*x^9*y - 9784775/8192*x^7*y + 2739737/16384*x^5*y - 85085/8192*x^3*y + 12155/65536*x*y
18
2:
347123925225/32768*x^16*y^2 - 148767396525/4096*x^14*y^2 + 410237366175/8192*x^12*y^2 - 145568097675/4096*x^10*y^2 + 225881530875/16384*x^8*y^2 - 11712375675/4096*x^6*y^2 + 2342475135/8192*x^4*y^2 - 43648605/4096*x^2*y^2 + 2078505/32768*y^2
18
-2:
23881935/262144*x^16*y^2 - 10235115/32768*x^14*y^2 + 28224105/65536*x^12*y^2 - 10015005/32768*x^10*y^2 + 15540525/131072*x^8*y^2 - 805805/32768*x^6*y^2 + 161161/65536*x^4*y^2 - 3003/32768*x^2*y^2 + 143/262144*y^2
18
3:
-347123925225/2048*x^15*y^3 + 1041371775675/2048*x^13*y^3 - 1230712098525/2048*x^11*y^3 + 727840488375/2048*x^9*y^3 - 225881530875/2048*x^7*y^3 + 35137127025/2048*x^5*y^3 - 2342475135/2048*x^3*y^3 + 43648605/2048*x*y^3
18
-3:
1137235/262144*x^15*y^3 - 3411705/262144*x^13*y^3 + 4032015/262144*x^11*y^3 - 2384525/262144*x^9*y^3 + 740025/262144*x^7*y^3 - 115115/262144*x^5*y^3 + 23023/786432*x^3*y^3 - 143/262144*x*y^3
18
4:
5206858878375/2048*x^14*y^4 - 13537833083775/2048*x^12*y^4 + 13537833083775/2048*x^10*y^4 - 6550564395375/2048*x^8*y^4 + 1581170716125/2048*x^6*y^4 - 175685635125/2048*x^4*y^4 + 7027425405/2048*x^2*y^4 - 43648605/2048*y^4
18
-4:
103385/524288*x^14*y^4 - 268801/524288*x^12*y^4 + 268801/524288*x^10*y^4 - 130065/524288*x^8*y^4 + 31395/524288*x^6*y^4 - 10465/1572864*x^4*y^4 + 2093/7864320*x^2*y^4 - 13/7864320*y^4
18
5:
-36448012148625/1024*x^13*y^5 + 40613499251325/512*x^11*y^5 - 67689165418875/1024*x^9*y^5 + 6550564395375/256*x^7*y^5 - 4743512148375/1024*x^5*y^5 + 175685635125/512*x^3*y^5 - 7027425405/1024*x*y^5
18
-5:
4495/524288*x^13*y^5 - 35061/1835008*x^11*y^5 + 58435/3670016*x^9*y^5 - 5655/917504*x^7*y^5 + 585/524288*x^5*y^5 - 65/786432*x^3*y^5 + 13/7864320*x*y^5
18
6:
473824157932125/1024*x^12*y^6 - 446748491764575/512*x^10*y^6 + 609202488769875/1024*x^8*y^6 - 45853950767625/256*x^6*y^6 + 23717560741875/1024*x^4*y^6 - 527056905375/512*x^2*y^6 + 7027425405/1024*y^6
18
-6:
4495/12582912*x^12*y^6 - 9889/14680064*x^10*y^6 + 13485/29360128*x^8*y^6 - 145/1048576*x^6*y^6 + 75/4194304*x^4*y^6 - 5/6291456*x^2*y^6 + 1/188743680*y^6
18
7:
-1421472473796375/256*x^11*y^7 + 2233742458822875/256*x^9*y^7 - 609202488769875/128*x^7*y^7 + 137561852302875/128*x^5*y^7 - 23717560741875/256*x^3*y^7 + 527056905375/256*x*y^7
18
-7:
899/62914560*x^11*y^7 - 9889/440401920*x^9*y^7 + 899/73400320*x^7*y^7 - 29/10485760*x^5*y^7 + 1/4194304*x^3*y^7 - 1/188743680*x*y^7
18
8:
15636197211760125/256*x^10*y^8 - 20103682129405875/256*x^8*y^8 + 4264417421389125/128*x^6*y^8 - 687809261514375/128*x^4*y^8 + 71152682225625/256*x^2*y^8 - 527056905375/256*y^8
18
-8:
899/1635778560*x^10*y^8 - 2697/3816816640*x^8*y^8 + 899/2998927360*x^6*y^8 - 29/599785472*x^4*y^8 + 3/1199570944*x^2*y^8 - 1/53980692480*y^8
18
9:
-78180986058800625/128*x^9*y^9 + 20103682129405875/32*x^7*y^9 - 12793252264167375/64*x^5*y^9 + 687809261514375/32*x^3*y^9 - 71152682225625/128*x*y^9
18
-9:
899/44166021120*x^9*y^9 - 899/42939187200*x^7*y^9 + 899/134951731200*x^5*y^9 - 29/40485519360*x^3*y^9 + 1/53980692480*x*y^9
18
10:
703628874529205625/128*x^8*y^10 - 140725774905841125/32*x^6*y^10 + 63966261320836875/64*x^4*y^10 - 2063427784543125/32*x^2*y^10 + 71152682225625/128*y^10
18
-10:
899/1236648591360*x^8*y^10 - 899/1545810739200*x^6*y^10 + 899/6801567252480*x^4*y^10 - 29/3400783626240*x^2*y^10 + 1/13603134504960*y^10
18
11:
-703628874529205625/16*x^7*y^11 + 422177324717523375/16*x^5*y^11 - 63966261320836875/16*x^3*y^11 + 2063427784543125/16*x*y^11
18
-11:
31/1236648591360*x^7*y^11 - 31/2061080985600*x^5*y^11 + 31/13603134504960*x^3*y^11 - 1/13603134504960*x*y^11
18
12:
4925402121704439375/16*x^6*y^12 - 2110886623587616875/16*x^4*y^12 + 191898783962510625/16*x^2*y^12 - 2063427784543125/16*y^12
18
-12:
31/37099457740800*x^6*y^12 - 31/86565401395200*x^4*y^12 + 31/952219415347200*x^2*y^12 - 1/2856658246041600*y^12
18
13:
-14776206365113318125/8*x^5*y^13 + 2110886623587616875/4*x^3*y^13 - 191898783962510625/8*x*y^13
18
-13:
1/37099457740800*x^5*y^13 - 1/129848102092800*x^3*y^13 + 1/2856658246041600*x*y^13
18
14:
73881031825566590625/8*x^4*y^14 - 6332659870762850625/4*x^2*y^14 + 191898783962510625/8*y^14
18
-14:
1/1187182647705600*x^4*y^14 - 1/6925232111616000*x^2*y^14 + 1/457065319366656000*y^14
18
15:
-73881031825566590625/2*x^3*y^15 + 6332659870762850625/2*x*y^15
18
-15:
1/39177027374284800*x^3*y^15 - 1/457065319366656000*x*y^15
18
16:
221643095476699771875/2*x^2*y^16 - 6332659870762850625/2*y^16
18
-16:
1/1332018930725683200*x^2*y^16 - 1/46620662575398912000*y^16
18
17:
-221643095476699771875*x*y^17
18
-17:
1/46620662575398912000*x*y^17
18
18:
221643095476699771875*y^18
18
-18:
1/1678343852714360832000*y^18
19
0:
4418157975/65536*x^19 - 20419054425/65536*x^17 + 9917826435/16384*x^15 - 10518906825/16384*x^13 + 13233463425/32768*x^11 - 5019589575/32768*x^9 + 557732175/16384*x^7 - 66927861/16384*x^5 + 14549535/65536*x^3 - 230945/65536*x
19
1:
-83945001525/65536*x^18*y + 347123925225/65536*x^16*y - 148767396525/16384*x^14*y + 136745788725/16384*x^12*y - 145568097675/32768*x^10*y + 45176306175/32768*x^8*y - 3904125225/16384*x^6*y + 334639305/16384*x^4*y - 43648605/65536*x^2*y + 230945/65536*y
19
-1:
883631595/262144*x^18*y - 3653936055/262144*x^16*y + 1565972595/65536*x^14*y - 1439429355/65536*x^12*y + 1532295765/131072*x^10*y - 475540065/131072*x^8*y + 41096055/65536*x^6*y - 3522519/65536*x^4*y + 459459/262144*x^2*y - 2431/262144*y
19
2:
755505013725/32768*x^17*y^2 - 347123925225/4096*x^15*y^2 + 1041371775675/8192*x^13*y^2 - 410237366175/4096*x^11*y^2 + 727840488375/16384*x^9*y^2 - 45176306175/4096*x^7*y^2 + 11712375675/8192*x^5*y^2 - 334639305/4096*x^3*y^2 + 43648605/32768*x*y^2
19
-2:
42077695/262144*x^17*y^2 - 19332995/32768*x^15*y^2 + 57998985/65536*x^13*y^2 - 22848085/32768*x^11*y^2 + 40536925/131072*x^9*y^2 - 2516085/32768*x^7*y^2 + 1956955/196608*x^5*y^2 - 55913/98304*x^3*y^2 + 2431/262144*x*y^2
19
3:
-12843585233325/32768*x^16*y^3 + 5206858878375/4096*x^14*y^3 - 13537833083775/8192*x^12*y^3 + 4512611027925/4096*x^10*y^3 - 6550564395375/16384*x^8*y^3 + 316234143225/4096*x^6*y^3 - 58561878375/8192*x^4*y^3 + 1003917915/4096*x^2*y^3 - 43648605/32768*y^3
19
-3:
3825245/524288*x^16*y^3 - 1550775/65536*x^14*y^3 + 4032015/131072*x^12*y^3 - 1344005/65536*x^10*y^3 + 1950975/262144*x^8*y^3 - 94185/65536*x^6*y^3 + 52325/393216*x^4*y^3 - 299/65536*x^2*y^3 + 13/524288*y^3
19
4:
12843585233325/2048*x^15*y^4 - 36448012148625/2048*x^13*y^4 + 40613499251325/2048*x^11*y^4 - 22563055139625/2048*x^9*y^4 + 6550564395375/2048*x^7*y^4 - 948702429675/2048*x^5*y^4 + 58561878375/2048*x^3*y^4 - 1003917915/2048*x*y^4
19
-4:
166315/524288*x^15*y^4 - 471975/524288*x^13*y^4 + 525915/524288*x^11*y^4 - 292175/524288*x^9*y^4 + 84825/524288*x^7*y^4 - 12285/524288*x^5*y^4 + 2275/1572864*x^3*y^4 - 13/524288*x*y^4
19
5:
-192653778499875/2048*x^14*y^5 + 473824157932125/2048*x^12*y^5 - 446748491764575/2048*x^10*y^5 + 203067496256625/2048*x^8*y^5 - 45853950767625/2048*x^6*y^5 + 4743512148375/2048*x^4*y^5 - 175685635125/2048*x^2*y^5 + 1003917915/2048*y^5
19
-5:
166315/12582912*x^14*y^5 - 409045/12582912*x^12*y^5 + 128557/4194304*x^10*y^5 - 58435/4194304*x^8*y^5 + 13195/4194304*x^6*y^5 - 1365/4194304*x^4*y^5 + 455/37748736*x^2*y^5 - 13/188743680*y^5
19
6:
1348576449499125/1024*x^13*y^6 - 1421472473796375/512*x^11*y^6 + 2233742458822875/1024*x^9*y^6 - 203067496256625/256*x^7*y^6 + 137561852302875/1024*x^5*y^6 - 4743512148375/512*x^3*y^6 + 175685635125/1024*x*y^6
19
-6:
33263/62914560*x^13*y^6 - 11687/10485760*x^11*y^6 + 128557/146800640*x^9*y^6 - 11687/36700160*x^7*y^6 + 1131/20971520*x^5*y^6 - 39/10485760*x^3*y^6 + 13/188743680*x*y^6
19
7:
-17531493843488625/1024*x^12*y^7 + 15636197211760125/512*x^10*y^7 - 20103682129405875/1024*x^8*y^7 + 1421472473796375/256*x^6*y^7 - 687809261514375/1024*x^4*y^7 + 14230536445125/512*x^2*y^7 - 175685635125/1024*y^7
19
-7:
33263/1635778560*x^12*y^7 - 9889/272629760*x^10*y^7 + 89001/3816816640*x^8*y^7 - 899/136314880*x^6*y^7 + 87/109051904*x^4*y^7 - 9/272629760*x^2*y^7 + 1/4907335680*y^7
19
8:
52594481530465875/256*x^11*y^8 - 78180986058800625/256*x^9*y^8 + 20103682129405875/128*x^7*y^8 - 4264417421389125/128*x^5*y^8 + 687809261514375/256*x^3*y^8 - 14230536445125/256*x*y^8
19
-8:
33263/44166021120*x^11*y^8 - 9889/8833204224*x^9*y^8 + 9889/17175674880*x^7*y^8 - 899/7361003520*x^5*y^8 + 29/2944401408*x^3*y^8 - 1/4907335680*x*y^8
19
9:
-578539296835124625/256*x^10*y^9 + 703628874529205625/256*x^8*y^9 - 140725774905841125/128*x^6*y^9 + 21322087106945625/128*x^4*y^9 - 2063427784543125/256*x^2*y^9 + 14230536445125/256*y^9
19
-9:
33263/1236648591360*x^10*y^9 - 899/27481079808*x^8*y^9 + 899/68702699520*x^6*y^9 - 899/453437816832*x^4*y^9 + 29/302291877888*x^2*y^9 - 1/1511459389440*y^9
19
10:
2892696484175623125/128*x^9*y^10 - 703628874529205625/32*x^7*y^10 + 422177324717523375/64*x^5*y^10 - 21322087106945625/32*x^3*y^10 + 2063427784543125/128*x*y^10
19
-10:
1147/1236648591360*x^9*y^10 - 31/34351349760*x^7*y^10 + 31/114504499200*x^5*y^10 - 31/1133594542080*x^3*y^10 + 1/1511459389440*x*y^10
19
11:
-26034268357580608125/128*x^8*y^11 + 4925402121704439375/32*x^6*y^11 - 2110886623587616875/64*x^4*y^11 + 63966261320836875/32*x^2*y^11 - 2063427784543125/128*y^11
19
-11:
1147/37099457740800*x^8*y^11 - 31/1324980633600*x^6*y^11 + 31/6183242956800*x^4*y^11 - 31/102023508787200*x^2*y^11 + 1/408094035148800*y^11
19
12:
26034268357580608125/16*x^7*y^12 - 14776206365113318125/16*x^5*y^12 + 2110886623587616875/16*x^3*y^12 - 63966261320836875/16*x*y^12
19
-12:
37/37099457740800*x^7*y^12 - 1/1766640844800*x^5*y^12 + 1/12366485913600*x^3*y^12 - 1/408094035148800*x*y^12
19
13:
-182239878503064256875/16*x^6*y^13 + 73881031825566590625/16*x^4*y^13 - 6332659870762850625/16*x^2*y^13 + 63966261320836875/16*y^13
19
-13:
37/1187182647705600*x^6*y^13 - 1/79145509847040*x^4*y^13 + 1/923364281548800*x^2*y^13 - 1/91413063873331200*y^13
19
14:
546719635509192770625/8*x^5*y^14 - 73881031825566590625/4*x^3*y^14 + 6332659870762850625/8*x*y^14
19
-14:
37/39177027374284800*x^5*y^14 - 1/3917702737428480*x^3*y^14 + 1/91413063873331200*x*y^14
19
15:
-2733598177545963853125/8*x^4*y^15 + 221643095476699771875/4*x^2*y^15 - 6332659870762850625/8*y^15
19
-15:
37/1332018930725683200*x^4*y^15 - 1/222003155120947200*x^2*y^15 + 1/15540220858466304000*y^15
19
16:
2733598177545963853125/2*x^3*y^16 - 221643095476699771875/2*x*y^16
19
-16:
37/46620662575398912000*x^3*y^16 - 1/15540220858466304000*x*y^16
19
17:
-8200794532637891559375/2*x^2*y^17 + 221643095476699771875/2*y^17
19
-17:
37/1678343852714360832000*x^2*y^17 - 1/1678343852714360832000*y^17
19
18:
8200794532637891559375*x*y^18
19
-18:
1/1678343852714360832000*x*y^18
19
19:
-8200794532637891559375*y^19
19
-19:
1/63777066403145711616000*y^19
20
0:
34461632205/262144*x^20 - 83945001525/131072*x^18 + 347123925225/262144*x^16 - 49589132175/32768*x^14 + 136745788725/131072*x^12 - 29113619535/65536*x^10 + 15058768725/131072*x^8 - 557732175/32768*x^6 + 334639305/262144*x^4 - 4849845/131072*x^2 + 46189/262144
20
1:
-172308161025/65536*x^19*y + 755505013725/65536*x^17*y - 347123925225/16384*x^15*y + 347123925225/16384*x^13*y - 410237366175/32768*x^11*y + 145568097675/32768*x^9*y - 15058768725/16384*x^7*y + 1673196525/16384*x^5*y - 334639305/65536*x^3*y + 4849845/65536*x*y
20
-1:
1641030105/262144*x^19*y - 7195285845/262144*x^17*y + 3305942145/65536*x^15*y - 3305942145/65536*x^13*y + 3907022535/131072*x^11*y - 1386362835/131072*x^9*y + 143416845/65536*x^7*y - 15935205/65536*x^5*y + 3187041/262144*x^3*y - 46189/262144*x*y
20
2:
3273855059475/65536*x^18*y^2 - 12843585233325/65536*x^16*y^2 + 5206858878375/16384*x^14*y^2 - 4512611027925/16384*x^12*y^2 + 4512611027925/32768*x^10*y^2 - 1310112879075/32768*x^8*y^2 + 105411381075/16384*x^6*y^2 - 8365982625/16384*x^4*y^2 + 1003917915/65536*x^2*y^2 - 4849845/65536*y^2
20
-2:
149184555/524288*x^18*y^2 - 585262485/524288*x^16*y^2 + 237268575/131072*x^14*y^2 - 205632765/131072*x^12*y^2 + 205632765/262144*x^10*y^2 - 59699835/262144*x^8*y^2 + 4803435/131072*x^6*y^2 - 381225/131072*x^4*y^2 + 45747/524288*x^2*y^2 - 221/524288*y^2
20
3:
-29464695535275/32768*x^17*y^3 + 12843585233325/4096*x^15*y^3 - 36448012148625/8192*x^13*y^3 + 13537833083775/4096*x^11*y^3 - 22563055139625/16384*x^9*y^3 + 1310112879075/4096*x^7*y^3 - 316234143225/8192*x^5*y^3 + 8365982625/4096*x^3*y^3 - 1003917915/32768*x*y^3
20
-3:
6486285/524288*x^17*y^3 - 2827355/65536*x^15*y^3 + 8023575/131072*x^13*y^3 - 2980185/65536*x^11*y^3 + 4966975/262144*x^9*y^3 - 288405/65536*x^7*y^3 + 69615/131072*x^5*y^3 - 5525/196608*x^3*y^3 + 221/524288*x*y^3
20
4:
500899824099675/32768*x^16*y^4 - 192653778499875/4096*x^14*y^4 + 473824157932125/8192*x^12*y^4 - 148916163921525/4096*x^10*y^4 + 203067496256625/16384*x^8*y^4 - 9170790153525/4096*x^6*y^4 + 1581170716125/8192*x^4*y^4 - 25097947875/4096*x^2*y^4 + 1003917915/32768*y^4
20
-4:
2162095/4194304*x^16*y^4 - 831575/524288*x^14*y^4 + 2045225/1048576*x^12*y^4 - 642785/524288*x^10*y^4 + 876525/2097152*x^8*y^4 - 39585/524288*x^6*y^4 + 6825/1048576*x^4*y^4 - 325/1572864*x^2*y^4 + 13/12582912*y^4
20
5:
-500899824099675/2048*x^15*y^5 + 1348576449499125/2048*x^13*y^5 - 1421472473796375/2048*x^11*y^5 + 744580819607625/2048*x^9*y^5 - 203067496256625/2048*x^7*y^5 + 27512370460575/2048*x^5*y^5 - 1581170716125/2048*x^3*y^5 + 25097947875/2048*x*y^5
20
-5:
432419/20971520*x^15*y^5 - 232841/4194304*x^13*y^5 + 245427/4194304*x^11*y^5 - 128557/4194304*x^9*y^5 + 35061/4194304*x^7*y^5 - 23751/20971520*x^5*y^5 + 273/4194304*x^3*y^5 - 13/12582912*x*y^5
20
6:
7513497361495125/2048*x^14*y^6 - 17531493843488625/2048*x^12*y^6 + 15636197211760125/2048*x^10*y^6 - 6701227376468625/2048*x^8*y^6 + 1421472473796375/2048*x^6*y^6 - 137561852302875/2048*x^4*y^6 + 4743512148375/2048*x^2*y^6 - 25097947875/2048*y^6
20
-6:
33263/41943040*x^14*y^6 - 232841/125829120*x^12*y^6 + 69223/41943040*x^10*y^6 - 29667/41943040*x^8*y^6 + 6293/41943040*x^6*y^6 - 609/41943040*x^4*y^6 + 21/41943040*x^2*y^6 - 1/377487360*y^6
20
7:
-52594481530465875/1024*x^13*y^7 + 52594481530465875/512*x^11*y^7 - 78180986058800625/1024*x^9*y^7 + 6701227376468625/256*x^7*y^7 - 4264417421389125/1024*x^5*y^7 + 137561852302875/512*x^3*y^7 - 4743512148375/1024*x*y^7
20
-7:
33263/1132462080*x^13*y^7 - 33263/566231040*x^11*y^7 + 9889/226492416*x^9*y^7 - 9889/660602880*x^7*y^7 + 899/377487360*x^5*y^7 - 29/188743680*x^3*y^7 + 1/377487360*x*y^7
20
8:
683728259896056375/1024*x^12*y^8 - 578539296835124625/512*x^10*y^8 + 703628874529205625/1024*x^8*y^8 - 46908591635280375/256*x^6*y^8 + 21322087106945625/1024*x^4*y^8 - 412685556908625/512*x^2*y^8 + 4743512148375/1024*y^8
20
-8:
33263/31708938240*x^12*y^8 - 365893/206108098560*x^10*y^8 + 9889/9160359936*x^8*y^8 - 9889/34351349760*x^6*y^8 + 899/27481079808*x^4*y^8 - 29/22900899840*x^2*y^8 + 1/137405399040*y^8
20
9:
-2051184779688169125/256*x^11*y^9 + 2892696484175623125/256*x^9*y^9 - 703628874529205625/128*x^7*y^9 + 140725774905841125/128*x^5*y^9 - 21322087106945625/256*x^3*y^9 + 412685556908625/256*x*y^9
20
-9:
1147/31708938240*x^11*y^9 - 12617/247329718272*x^9*y^9 + 341/13740539904*x^7*y^9 - 341/68702699520*x^5*y^9 + 31/82443239424*x^3*y^9 - 1/137405399040*x*y^9
20
10:
22563032576569860375/256*x^10*y^10 - 26034268357580608125/256*x^8*y^10 + 4925402121704439375/128*x^6*y^10 - 703628874529205625/128*x^4*y^10 + 63966261320836875/256*x^2*y^10 - 412685556908625/256*y^10
20
-10:
1147/951268147200*x^10*y^10 - 1147/824432394240*x^8*y^10 + 31/58888028160*x^6*y^10 - 31/412216197120*x^4*y^10 + 31/9068756336640*x^2*y^10 - 1/45343781683200*y^10
20
11:
-112815162882849301875/128*x^9*y^11 + 26034268357580608125/32*x^7*y^11 - 14776206365113318125/64*x^5*y^11 + 703628874529205625/32*x^3*y^11 - 63966261320836875/128*x*y^11
20
-11:
37/951268147200*x^9*y^11 - 37/1030540492800*x^7*y^11 + 1/98146713600*x^5*y^11 - 1/1030540492800*x^3*y^11 + 1/45343781683200*x*y^11
20
12:
1015336465945643716875/128*x^8*y^12 - 182239878503064256875/32*x^6*y^12 + 73881031825566590625/64*x^4*y^12 - 2110886623587616875/32*x^2*y^12 + 63966261320836875/128*y^12
20
-12:
37/30440580710400*x^8*y^12 - 37/42399380275200*x^6*y^12 + 1/5653250703360*x^4*y^12 - 1/98931887308800*x^2*y^12 + 1/13059009124761600*y^12
20
13:
-1015336465945643716875/16*x^7*y^13 + 546719635509192770625/16*x^5*y^13 - 73881031825566590625/16*x^3*y^13 + 2110886623587616875/16*x*y^13
20
-13:
37/1004539163443200*x^7*y^13 - 37/1865572732108800*x^5*y^13 + 1/373114546421760*x^3*y^13 - 1/13059009124761600*x*y^13
20
14:
7107355261619506018125/16*x^6*y^14 - 2733598177545963853125/16*x^4*y^14 + 221643095476699771875/16*x^2*y^14 - 2110886623587616875/16*y^14
20
-14:
37/34154331557068800*x^6*y^14 - 37/88801262048378880*x^4*y^14 + 1/29600420682792960*x^2*y^14 - 1/3108044171693260800*y^14
20
15:
-21322065784858518054375/8*x^5*y^15 + 2733598177545963853125/4*x^3*y^15 - 221643095476699771875/8*x*y^15
20
-15:
37/1195401604497408000*x^5*y^15 - 37/4662066257539891200*x^3*y^15 + 1/3108044171693260800*x*y^15
20
16:
106610328924292590271875/8*x^4*y^16 - 8200794532637891559375/4*x^2*y^16 + 221643095476699771875/8*y^16
20
-16:
37/43034457761906688000*x^4*y^16 - 37/279723975452393472000*x^2*y^16 + 1/559447950904786944000*y^16
20
17:
-106610328924292590271875/2*x^3*y^17 + 8200794532637891559375/2*x*y^17
20
-17:
1/43034457761906688000*x^3*y^17 - 1/559447950904786944000*x*y^17
20
18:
319830986772877770815625/2*x^2*y^18 - 8200794532637891559375/2*y^18
20
-18:
1/1635309394952454144000*x^2*y^18 - 1/63777066403145711616000*y^18
20
19:
-319830986772877770815625*x*y^19
20
-19:
1/63777066403145711616000*x*y^19
20
20:
319830986772877770815625*y^20
20
-20:
1/2551082656125828464640000*y^20
21
0:
67282234305/262144*x^21 - 172308161025/131072*x^19 + 755505013725/262144*x^17 - 115707975075/32768*x^15 + 347123925225/131072*x^13 - 82047473235/65536*x^11 + 48522699225/131072*x^9 - 2151252675/32768*x^7 + 1673196525/262144*x^5 - 37182145/131072*x^3 + 969969/262144*x
21
1:
-1412926920405/262144*x^20*y + 3273855059475/131072*x^18*y - 12843585233325/262144*x^16*y + 1735619626125/32768*x^14*y - 4512611027925/131072*x^12*y + 902522205585/65536*x^10*y - 436704293025/131072*x^8*y + 15058768725/32768*x^6*y - 8365982625/262144*x^4*y + 111546435/131072*x^2*y - 969969/262144*y
21
-1:
6116566755/524288*x^20*y - 14172532725/262144*x^18*y + 55599936075/524288*x^16*y - 7513504875/65536*x^14*y + 19535112675/262144*x^12*y - 3907022535/131072*x^10*y + 1890494775/262144*x^8*y - 65189475/65536*x^6*y + 36216375/524288*x^4*y - 482885/262144*x^2*y + 4199/524288*y
21
2:
7064634602025/65536*x^19*y^2 - 29464695535275/65536*x^17*y^2 + 12843585233325/16384*x^15*y^2 - 12149337382875/16384*x^13*y^2 + 13537833083775/32768*x^11*y^2 - 4512611027925/32768*x^9*y^2 + 436704293025/16384*x^7*y^2 - 45176306175/16384*x^5*y^2 + 8365982625/65536*x^3*y^2 - 111546435/65536*x*y^2
21
-2:
265937685/524288*x^19*y^2 - 1109154735/524288*x^17*y^2 + 483477705/131072*x^15*y^2 - 457343775/131072*x^13*y^2 + 509611635/262144*x^11*y^2 - 169870545/262144*x^9*y^2 + 16439085/131072*x^7*y^2 - 1700595/131072*x^5*y^2 + 314925/524288*x^3*y^2 - 4199/524288*x*y^2
21
3:
-134228057438475/65536*x^18*y^3 + 500899824099675/65536*x^16*y^3 - 192653778499875/16384*x^14*y^3 + 157941385977375/16384*x^12*y^3 - 148916163921525/32768*x^10*y^3 + 40613499251325/32768*x^8*y^3 - 3056930051175/16384*x^6*y^3 + 225881530875/16384*x^4*y^3 - 25097947875/65536*x^2*y^3 + 111546435/65536*y^3
21
-3:
88645895/4194304*x^18*y^3 - 330800535/4194304*x^16*y^3 + 127230975/1048576*x^14*y^3 - 104306475/1048576*x^12*y^3 + 98346105/2097152*x^10*y^3 - 26821665/2097152*x^8*y^3 + 2018835/1048576*x^6*y^3 - 149175/1048576*x^4*y^3 + 16575/4194304*x^2*y^3 - 221/12582912*y^3
21
4:
1208052516946275/32768*x^17*y^4 - 500899824099675/4096*x^15*y^4 + 1348576449499125/8192*x^13*y^4 - 473824157932125/4096*x^11*y^4 + 744580819607625/16384*x^9*y^4 - 40613499251325/4096*x^7*y^4 + 9170790153525/8192*x^5*y^4 - 225881530875/4096*x^3*y^4 + 25097947875/32768*x*y^4
21
-4:
17729179/20971520*x^17*y^4 - 7351123/2621440*x^15*y^4 + 3958297/1048576*x^13*y^4 - 1390753/524288*x^11*y^4 + 2185469/2097152*x^9*y^4 - 596037/2621440*x^7*y^4 + 134589/5242880*x^5*y^4 - 663/524288*x^3*y^4 + 221/12582912*x*y^4
21
5:
-20536892788086675/32768*x^16*y^5 + 7513497361495125/4096*x^14*y^5 - 17531493843488625/8192*x^12*y^5 + 5212065737253375/4096*x^10*y^5 - 6701227376468625/16384*x^8*y^5 + 284294494759275/4096*x^6*y^5 - 45853950767625/8192*x^4*y^5 + 677644592625/4096*x^2*y^5 - 25097947875/32768*y^5
21
-5:
1363783/41943040*x^16*y^5 - 99789/1048576*x^14*y^5 + 232841/2097152*x^12*y^5 - 69223/1048576*x^10*y^5 + 89001/4194304*x^8*y^5 - 18879/5242880*x^6*y^5 + 609/2097152*x^4*y^5 - 9/1048576*x^2*y^5 + 1/25165824*y^5
21
6:
20536892788086675/2048*x^15*y^6 - 52594481530465875/2048*x^13*y^6 + 52594481530465875/2048*x^11*y^6 - 26060328686266875/2048*x^9*y^6 + 6701227376468625/2048*x^7*y^6 - 852883484277825/2048*x^5*y^6 + 45853950767625/2048*x^3*y^6 - 677644592625/2048*x*y^6
21
-6:
1363783/1132462080*x^15*y^6 - 232841/75497472*x^13*y^6 + 232841/75497472*x^11*y^6 - 346115/226492416*x^9*y^6 + 9889/25165824*x^7*y^6 - 6293/125829120*x^5*y^6 + 203/75497472*x^3*y^6 - 1/25165824*x*y^6
21
7:
-308053391821300125/2048*x^14*y^7 + 683728259896056375/2048*x^12*y^7 - 578539296835124625/2048*x^10*y^7 + 234542958176401875/2048*x^8*y^7 - 46908591635280375/2048*x^6*y^7 + 4264417421389125/2048*x^4*y^7 - 137561852302875/2048*x^2*y^7 + 677644592625/2048*y^7
21
-7:
1363783/31708938240*x^14*y^7 - 432419/4529848320*x^12*y^7 + 365893/4529848320*x^10*y^7 - 9889/301989888*x^8*y^7 + 9889/1509949440*x^6*y^7 - 899/1509949440*x^4*y^7 + 29/1509949440*x^2*y^7 - 1/10569646080*y^7
21
8:
2156373742749100875/1024*x^13*y^8 - 2051184779688169125/512*x^11*y^8 + 2892696484175623125/1024*x^9*y^8 - 234542958176401875/256*x^7*y^8 + 140725774905841125/1024*x^5*y^8 - 4264417421389125/512*x^3*y^8 + 137561852302875/1024*x*y^8
21
-8:
47027/31708938240*x^13*y^8 - 14911/5284823040*x^11*y^8 + 12617/6341787648*x^9*y^8 - 341/528482304*x^7*y^8 + 341/3523215360*x^5*y^8 - 31/5284823040*x^3*y^8 + 1/10569646080*x*y^8
21
9:
-28032858655738311375/1024*x^12*y^9 + 22563032576569860375/512*x^10*y^9 - 26034268357580608125/1024*x^8*y^9 + 1641800707234813125/256*x^6*y^9 - 703628874529205625/1024*x^4*y^9 + 12793252264167375/512*x^2*y^9 - 137561852302875/1024*y^9
21
-9:
47027/951268147200*x^12*y^9 - 12617/158544691200*x^10*y^9 + 12617/274810798080*x^8*y^9 - 341/29444014080*x^6*y^9 + 341/274810798080*x^4*y^9 - 31/687026995200*x^2*y^9 + 1/4122161971200*y^9
21
10:
84098575967214934125/256*x^11*y^10 - 112815162882849301875/256*x^9*y^10 + 26034268357580608125/128*x^7*y^10 - 4925402121704439375/128*x^5*y^10 + 703628874529205625/256*x^3*y^10 - 12793252264167375/256*x*y^10
21
-10:
1517/951268147200*x^11*y^10 - 407/190253629440*x^9*y^10 + 407/412216197120*x^7*y^10 - 11/58888028160*x^5*y^10 + 11/824432394240*x^3*y^10 - 1/4122161971200*x*y^10
21
11:
-925084335639364275375/256*x^10*y^11 + 1015336465945643716875/256*x^8*y^11 - 182239878503064256875/128*x^6*y^11 + 24627010608522196875/128*x^4*y^11 - 2110886623587616875/256*x^2*y^11 + 12793252264167375/256*y^11
21
-11:
1517/30440580710400*x^10*y^11 - 37/676457349120*x^8*y^11 + 37/1884416901120*x^6*y^11 - 1/376883380224*x^4*y^11 + 1/8793945538560*x^2*y^11 - 1/1451001013862400*y^11
21
12:
4625421678196821376875/128*x^9*y^12 - 1015336465945643716875/32*x^7*y^12 + 546719635509192770625/64*x^5*y^12 - 24627010608522196875/32*x^3*y^12 + 2110886623587616875/128*x*y^12
21
-12:
1517/1004539163443200*x^9*y^12 - 37/27903865651200*x^7*y^12 + 37/103642929561600*x^5*y^12 - 1/31092878868480*x^3*y^12 + 1/1451001013862400*x*y^12
21
13:
-41628795103771392391875/128*x^8*y^13 + 7107355261619506018125/32*x^6*y^13 - 2733598177545963853125/64*x^4*y^13 + 73881031825566590625/32*x^2*y^13 - 2110886623587616875/128*y^13
21
-13:
1517/34154331557068800*x^8*y^13 - 37/1219797555609600*x^6*y^13 + 37/6342947289169920*x^4*y^13 - 1/3171473644584960*x^2*y^13 + 1/444006310241894400*y^13
21
14:
41628795103771392391875/16*x^7*y^14 - 21322065784858518054375/16*x^5*y^14 + 2733598177545963853125/16*x^3*y^14 - 73881031825566590625/16*x*y^14
21
-14:
1517/1195401604497408000*x^7*y^14 - 37/56923885928448000*x^5*y^14 + 37/444006310241894400*x^3*y^14 - 1/444006310241894400*x*y^14
21
15:
-291401565726399746743125/16*x^6*y^15 + 106610328924292590271875/16*x^4*y^15 - 8200794532637891559375/16*x^2*y^15 + 73881031825566590625/16*y^15
21
-15:
1517/43034457761906688000*x^6*y^15 - 37/2868963850793779200*x^4*y^15 + 37/37296530060319129600*x^2*y^15 - 1/111889590180957388800*y^15
21
16:
874204697179199240229375/8*x^5*y^16 - 106610328924292590271875/4*x^3*y^16 + 8200794532637891559375/8*x*y^16
21
-16:
41/43034457761906688000*x^5*y^16 - 1/4303445776190668800*x^3*y^16 + 1/111889590180957388800*x*y^16
21
17:
-4371023485895996201146875/8*x^4*y^17 + 319830986772877770815625/4*x^2*y^17 - 8200794532637891559375/8*y^17
21
-17:
41/1635309394952454144000*x^4*y^17 - 1/272551565825409024000*x^2*y^17 + 1/21259022134381903872000*y^17
21
18:
4371023485895996201146875/2*x^3*y^18 - 319830986772877770815625/2*x*y^18
21
-18:
41/63777066403145711616000*x^3*y^18 - 1/21259022134381903872000*x*y^18
21
19:
-13113070457687988603440625/2*x^2*y^19 + 319830986772877770815625/2*y^19
21
-19:
41/2551082656125828464640000*x^2*y^19 - 1/2551082656125828464640000*y^19
21
20:
13113070457687988603440625*x*y^20
21
-20:
1/2551082656125828464640000*x*y^20
21
21:
-13113070457687988603440625*y^21
21
-21:
1/107145471557284795514880000*y^21
22
0:
263012370465/524288*x^22 - 1412926920405/524288*x^20 + 3273855059475/524288*x^18 - 4281195077775/524288*x^16 + 1735619626125/262144*x^14 - 902522205585/262144*x^12 + 300840735195/262144*x^10 - 62386327575/262144*x^8 + 15058768725/524288*x^6 - 929553625/524288*x^4 + 22309287/524288*x^2 - 88179/524288
22
1:
-2893136075115/262144*x^21*y + 7064634602025/131072*x^19*y - 29464695535275/262144*x^17*y + 4281195077775/32768*x^15*y - 12149337382875/131072*x^13*y + 2707566616755/65536*x^11*y - 1504203675975/131072*x^9*y + 62386327575/32768*x^7*y - 45176306175/262144*x^5*y + 929553625/131072*x^3*y - 22309287/262144*x*y
22
-1:
11435320455/524288*x^21*y - 27923456925/262144*x^19*y + 116461247175/524288*x^17*y - 16921719675/65536*x^15*y + 48021096375/262144*x^13*y - 10701844335/131072*x^11*y + 5945469075/262144*x^9*y - 246586275/65536*x^7*y + 178562475/524288*x^5*y - 3674125/262144*x^3*y + 88179/524288*x*y
22
2:
60755857577415/262144*x^20*y^2 - 134228057438475/131072*x^18*y^2 + 500899824099675/262144*x^16*y^2 - 64217926166625/32768*x^14*y^2 + 157941385977375/131072*x^12*y^2 - 29783232784305/65536*x^10*y^2 + 13537833083775/131072*x^8*y^2 - 436704293025/32768*x^6*y^2 + 225881530875/262144*x^4*y^2 - 2788660875/131072*x^2*y^2 + 22309287/262144*y^2
22
-2:
3811773485/4194304*x^20*y^2 - 8421360025/2097152*x^18*y^2 + 31426050825/4194304*x^16*y^2 - 4028980875/524288*x^14*y^2 + 9909115125/2097152*x^12*y^2 - 1868575995/1048576*x^10*y^2 + 849352725/2097152*x^8*y^2 - 27398475/524288*x^6*y^2 + 14171625/4194304*x^4*y^2 - 524875/6291456*x^2*y^2 + 4199/12582912*y^2
22
3:
-303779287887075/65536*x^19*y^3 + 1208052516946275/65536*x^17*y^3 - 500899824099675/16384*x^15*y^3 + 449525483166375/16384*x^13*y^3 - 473824157932125/32768*x^11*y^3 + 148916163921525/32768*x^9*y^3 - 13537833083775/16384*x^7*y^3 + 1310112879075/16384*x^5*y^3 - 225881530875/65536*x^3*y^3 + 2788660875/65536*x*y^3
22
-3:
762354697/20971520*x^19*y^3 - 3031689609/20971520*x^17*y^3 + 1257042033/5242880*x^15*y^3 - 225622929/1048576*x^13*y^3 + 237818763/2097152*x^11*y^3 - 373715199/10485760*x^9*y^3 + 33974109/5242880*x^7*y^3 - 3287817/5242880*x^5*y^3 + 113373/4194304*x^3*y^3 - 4199/12582912*x*y^3
22
4:
5771806469854425/65536*x^18*y^4 - 20536892788086675/65536*x^16*y^4 + 7513497361495125/16384*x^14*y^4 - 5843831281162875/16384*x^12*y^4 + 5212065737253375/32768*x^10*y^4 - 1340245475293725/32768*x^8*y^4 + 94764831586425/16384*x^6*y^4 - 6550564395375/16384*x^4*y^4 + 677644592625/65536*x^2*y^4 - 2788660875/65536*y^4
22
-4:
58642669/41943040*x^18*y^4 - 208658799/41943040*x^16*y^4 + 15267717/2097152*x^14*y^4 - 11874891/2097152*x^12*y^4 + 10591119/4194304*x^10*y^4 - 13617153/20971520*x^8*y^4 + 962829/10485760*x^6*y^4 - 13311/2097152*x^4*y^4 + 1377/8388608*x^2*y^4 - 17/25165824*y^4
22
5:
-51946258228689825/32768*x^17*y^5 + 20536892788086675/4096*x^15*y^5 - 52594481530465875/8192*x^13*y^5 + 17531493843488625/4096*x^11*y^5 - 26060328686266875/16384*x^9*y^5 + 1340245475293725/4096*x^7*y^5 - 284294494759275/8192*x^5*y^5 + 6550564395375/4096*x^3*y^5 - 677644592625/32768*x*y^5
22
-5:
58642669/1132462080*x^17*y^5 - 23184311/141557760*x^15*y^5 + 3958297/18874368*x^13*y^5 - 3958297/28311552*x^11*y^5 + 5883955/113246208*x^9*y^5 - 168113/15728640*x^7*y^5 + 106981/94371840*x^5*y^5 - 493/9437184*x^3*y^5 + 17/25165824*x*y^5
22
6:
883086389887727025/32768*x^16*y^6 - 308053391821300125/4096*x^14*y^6 + 683728259896056375/8192*x^12*y^6 - 192846432278374875/4096*x^10*y^6 + 234542958176401875/16384*x^8*y^6 - 9381718327056075/4096*x^6*y^6 + 1421472473796375/8192*x^4*y^6 - 19651693186125/4096*x^2*y^6 + 677644592625/32768*y^6
22
-6:
58642669/31708938240*x^16*y^6 - 1363783/264241152*x^14*y^6 + 432419/75497472*x^12*y^6 - 365893/113246208*x^10*y^6 + 49445/50331648*x^8*y^6 - 9889/62914560*x^6*y^6 + 899/75497472*x^4*y^6 - 29/88080384*x^2*y^6 + 1/704643072*y^6
22
7:
-883086389887727025/2048*x^15*y^7 + 2156373742749100875/2048*x^13*y^7 - 2051184779688169125/2048*x^11*y^7 + 964232161391874375/2048*x^9*y^7 - 234542958176401875/2048*x^7*y^7 + 28145154981168225/2048*x^5*y^7 - 1421472473796375/2048*x^3*y^7 + 19651693186125/2048*x*y^7
22
-7:
2022161/31708938240*x^15*y^7 - 47027/301989888*x^13*y^7 + 14911/100663296*x^11*y^7 - 63085/905969664*x^9*y^7 + 1705/100663296*x^7*y^7 - 341/167772160*x^5*y^7 + 31/301989888*x^3*y^7 - 1/704643072*x*y^7
22
8:
13246295848315905375/2048*x^14*y^8 - 28032858655738311375/2048*x^12*y^8 + 22563032576569860375/2048*x^10*y^8 - 8678089452526869375/2048*x^8*y^8 + 1641800707234813125/2048*x^6*y^8 - 140725774905841125/2048*x^4*y^8 + 4264417421389125/2048*x^2*y^8 - 19651693186125/2048*y^8
22
-8:
2022161/951268147200*x^14*y^8 - 611351/135895449600*x^12*y^8 + 164021/45298483200*x^10*y^8 - 12617/9059696640*x^8*y^8 + 2387/9059696640*x^6*y^8 - 341/15099494400*x^4*y^8 + 31/45298483200*x^2*y^8 - 1/317089382400*y^8
22
9:
-92724070938211337625/1024*x^13*y^9 + 84098575967214934125/512*x^11*y^9 - 112815162882849301875/1024*x^9*y^9 + 8678089452526869375/256*x^7*y^9 - 4925402121704439375/1024*x^5*y^9 + 140725774905841125/512*x^3*y^9 - 4264417421389125/1024*x*y^9
22
-9:
65231/951268147200*x^13*y^9 - 19721/158544691200*x^11*y^9 + 5291/63417876480*x^9*y^9 - 407/15854469120*x^7*y^9 + 11/3019898880*x^5*y^9 - 11/52848230400*x^3*y^9 + 1/317089382400*x*y^9
22
10:
1205412922196747389125/1024*x^12*y^10 - 925084335639364275375/512*x^10*y^10 + 1015336465945643716875/1024*x^8*y^10 - 60746626167688085625/256*x^6*y^10 + 24627010608522196875/1024*x^4*y^10 - 422177324717523375/512*x^2*y^10 + 4264417421389125/1024*y^10
22
-10:
65231/30440580710400*x^12*y^10 - 16687/5073430118400*x^10*y^10 + 407/225485783040*x^8*y^10 - 407/942208450560*x^6*y^10 + 11/251255586816*x^4*y^10 - 11/7328287948800*x^2*y^10 + 1/131909183078400*y^10
22
11:
-3616238766590242167375/256*x^11*y^11 + 4625421678196821376875/256*x^9*y^11 - 1015336465945643716875/128*x^7*y^11 + 182239878503064256875/128*x^5*y^11 - 24627010608522196875/256*x^3*y^11 + 422177324717523375/256*x*y^11
22
-11:
65231/1004539163443200*x^11*y^11 - 1517/18264348426240*x^9*y^11 + 37/1014686023680*x^7*y^11 - 37/5653250703360*x^5*y^11 + 1/2261300281344*x^3*y^11 - 1/131909183078400*x*y^11
22
12:
39778626432492663841125/256*x^10*y^12 - 41628795103771392391875/256*x^8*y^12 + 7107355261619506018125/128*x^6*y^12 - 911199392515321284375/128*x^4*y^12 + 73881031825566590625/256*x^2*y^12 - 422177324717523375/256*y^12
22
-12:
65231/34154331557068800*x^10*y^12 - 1517/758985145712640*x^8*y^12 + 37/54213224693760*x^6*y^12 - 37/422863152611328*x^4*y^12 + 1/281908768407552*x^2*y^12 - 1/49334034471321600*y^12
22
13:
-198893132162463319205625/128*x^9*y^13 + 41628795103771392391875/32*x^7*y^13 - 21322065784858518054375/64*x^5*y^13 + 911199392515321284375/32*x^3*y^13 - 73881031825566590625/128*x*y^13
22
-13:
65231/1195401604497408000*x^9*y^13 - 1517/33205600124928000*x^7*y^13 + 37/3162438107136000*x^5*y^13 - 37/37000525853491200*x^3*y^13 + 1/49334034471321600*x*y^13
22
14:
1790038189462169872850625/128*x^8*y^14 - 291401565726399746743125/32*x^6*y^14 + 106610328924292590271875/64*x^4*y^14 - 2733598177545963853125/32*x^2*y^14 + 73881031825566590625/128*y^14
22
-14:
65231/43034457761906688000*x^8*y^14 - 1517/1536944920068096000*x^6*y^14 + 37/204925989342412800*x^4*y^14 - 37/3996056792177049600*x^2*y^14 + 1/15984227168708198400*y^14
22
15:
-1790038189462169872850625/16*x^7*y^15 + 874204697179199240229375/16*x^5*y^15 - 106610328924292590271875/16*x^3*y^15 + 2733598177545963853125/16*x*y^15
22
-15:
1763/43034457761906688000*x^7*y^15 - 41/2049259893424128000*x^5*y^15 + 1/409851978684825600*x^3*y^15 - 1/15984227168708198400*x*y^15
22
16:
12530267326235189109954375/16*x^6*y^16 - 4371023485895996201146875/16*x^4*y^16 + 319830986772877770815625/16*x^2*y^16 - 2733598177545963853125/16*y^16
22
-16:
1763/1635309394952454144000*x^6*y^16 - 41/109020626330163609600*x^4*y^16 + 1/36340208776721203200*x^2*y^16 - 1/4251804426876380774400*y^16
22
17:
-37590801978705567329863125/8*x^5*y^17 + 4371023485895996201146875/4*x^3*y^17 - 319830986772877770815625/8*x*y^17
22
-17:
1763/63777066403145711616000*x^5*y^17 - 41/6377706640314571161600*x^3*y^17 + 1/4251804426876380774400*x*y^17
22
18:
187954009893527836649315625/8*x^4*y^18 - 13113070457687988603440625/4*x^2*y^18 + 319830986772877770815625/8*y^18
22
-18:
1763/2551082656125828464640000*x^4*y^18 - 41/425180442687638077440000*x^2*y^18 + 1/850360885375276154880000*y^18
22
19:
-187954009893527836649315625/2*x^3*y^19 + 13113070457687988603440625/2*x*y^19
22
-19:
43/2551082656125828464640000*x^3*y^19 - 1/850360885375276154880000*x*y^19
22
20:
563862029680583509947946875/2*x^2*y^20 - 13113070457687988603440625/2*y^20
22
-20:
43/107145471557284795514880000*x^2*y^20 - 1/107145471557284795514880000*y^20
22
21:
-563862029680583509947946875*x*y^21
22
-21:
1/107145471557284795514880000*x*y^21
22
22:
563862029680583509947946875*y^22
22
-22:
1/4714400748520531002654720000*y^22
23
0:
514589420475/524288*x^23 - 2893136075115/524288*x^21 + 7064634602025/524288*x^19 - 9821565178425/524288*x^17 + 4281195077775/262144*x^15 - 2429867476575/262144*x^13 + 902522205585/262144*x^11 - 214886239425/262144*x^9 + 62386327575/524288*x^7 - 5019589575/524288*x^5 + 185910725/524288*x^3 - 2028117/524288*x
23
1:
-11835556670925/524288*x^22*y + 60755857577415/524288*x^20*y - 134228057438475/524288*x^18*y + 166966608033225/524288*x^16*y - 64217926166625/262144*x^14*y + 31588277195475/262144*x^12*y - 9927744261435/262144*x^10*y + 1933976154825/262144*x^8*y - 436704293025/524288*x^6*y + 25097947875/524288*x^4*y - 557732175/524288*x^2*y + 2028117/524288*y
23
-1:
171529806825/4194304*x^22*y - 880519675035/4194304*x^20*y + 1945334165775/4194304*x^18*y - 2419805913525/4194304*x^16*y + 930694582125/2097152*x^14*y - 457801118775/2097152*x^12*y + 143880351615/2097152*x^10*y - 28028639925/2097152*x^8*y + 6329047725/4194304*x^6*y - 363738375/4194304*x^4*y + 8083075/4194304*x^2*y - 29393/4194304*y
23
2:
130191123380175/262144*x^21*y^2 - 303779287887075/131072*x^19*y^2 + 1208052516946275/262144*x^17*y^2 - 166966608033225/32768*x^15*y^2 + 449525483166375/131072*x^13*y^2 - 94764831586425/65536*x^11*y^2 + 49638721307175/131072*x^9*y^2 - 1933976154825/32768*x^7*y^2 + 1310112879075/262144*x^5*y^2 - 25097947875/131072*x^3*y^2 + 557732175/262144*x*y^2
23
-2:
6861192273/4194304*x^21*y^2 - 16009448637/2097152*x^19*y^2 + 63665481789/4194304*x^17*y^2 - 8799294231/524288*x^15*y^2 + 23690407545/2097152*x^13*y^2 - 4994194023/1048576*x^11*y^2 + 2616006393/2097152*x^9*y^2 - 101922327/524288*x^7*y^2 + 69044157/4194304*x^5*y^2 - 1322685/2097152*x^3*y^2 + 29393/4194304*x*y^2
23
3:
-2734013590983675/262144*x^20*y^3 + 5771806469854425/131072*x^18*y^3 - 20536892788086675/262144*x^16*y^3 + 2504499120498375/32768*x^14*y^3 - 5843831281162875/131072*x^12*y^3 + 1042413147450675/65536*x^10*y^3 - 446748491764575/131072*x^8*y^3 + 13537833083775/32768*x^6*y^3 - 6550564395375/262144*x^4*y^3 + 75293843625/131072*x^2*y^3 - 557732175/262144*y^3
23
-3:
527784021/8388608*x^20*y^3 - 1114210711/4194304*x^18*y^3 + 3964517181/8388608*x^16*y^3 - 483477705/1048576*x^14*y^3 + 1128114645/4194304*x^12*y^3 - 201231261/2097152*x^10*y^3 + 86241969/4194304*x^8*y^3 - 2613393/1048576*x^6*y^3 + 1264545/8388608*x^4*y^3 - 14535/4194304*x^2*y^3 + 323/25165824*y^3
23
4:
13670067954918375/65536*x^19*y^4 - 51946258228689825/65536*x^17*y^4 + 20536892788086675/16384*x^15*y^4 - 17531493843488625/16384*x^13*y^4 + 17531493843488625/32768*x^11*y^4 - 5212065737253375/32768*x^9*y^4 + 446748491764575/16384*x^7*y^4 - 40613499251325/16384*x^5*y^4 + 6550564395375/65536*x^3*y^4 - 75293843625/65536*x*y^4
23
-4:
58642669/25165824*x^19*y^4 - 1114210711/125829120*x^17*y^4 + 440501909/31457280*x^15*y^4 - 75207643/6291456*x^13*y^4 + 75207643/12582912*x^11*y^4 - 22359029/12582912*x^9*y^4 + 3194147/10485760*x^7*y^4 - 290377/10485760*x^5*y^4 + 9367/8388608*x^3*y^4 - 323/25165824*x*y^4
23
5:
-259731291143449125/65536*x^18*y^5 + 883086389887727025/65536*x^16*y^5 - 308053391821300125/16384*x^14*y^5 + 227909419965352125/16384*x^12*y^5 - 192846432278374875/32768*x^10*y^5 + 46908591635280375/32768*x^8*y^5 - 3127239442352025/16384*x^6*y^5 + 203067496256625/16384*x^4*y^5 - 19651693186125/65536*x^2*y^5 + 75293843625/65536*y^5
23
-5:
58642669/704643072*x^18*y^5 - 996925373/3523215360*x^16*y^5 + 23184311/58720256*x^14*y^5 - 7351123/25165824*x^12*y^5 + 6220181/50331648*x^10*y^5 - 504339/16777216*x^8*y^5 + 168113/41943040*x^6*y^5 - 15283/58720256*x^4*y^5 + 1479/234881024*x^2*y^5 - 17/704643072*y^5
23
6:
2337581620291042125/32768*x^17*y^6 - 883086389887727025/4096*x^15*y^6 + 2156373742749100875/8192*x^13*y^6 - 683728259896056375/4096*x^11*y^6 + 964232161391874375/16384*x^9*y^6 - 46908591635280375/4096*x^7*y^6 + 9381718327056075/8192*x^5*y^6 - 203067496256625/4096*x^3*y^6 + 19651693186125/32768*x*y^6
23
-6:
2022161/704643072*x^17*y^6 - 34376737/3963617280*x^15*y^6 + 799459/75497472*x^13*y^6 - 253487/37748736*x^11*y^6 + 1072445/452984832*x^9*y^6 - 5797/12582912*x^7*y^6 + 5797/125829120*x^5*y^6 - 527/264241152*x^3*y^6 + 17/704643072*x*y^6
23
7:
-39738887544947716125/32768*x^16*y^7 + 13246295848315905375/4096*x^14*y^7 - 28032858655738311375/8192*x^12*y^7 + 7521010858856620125/4096*x^10*y^7 - 8678089452526869375/16384*x^8*y^7 + 328360141446962625/4096*x^6*y^7 - 46908591635280375/8192*x^4*y^7 + 609202488769875/4096*x^2*y^7 - 19651693186125/32768*y^7
23
-7:
2022161/21139292160*x^16*y^7 - 2022161/7927234560*x^14*y^7 + 611351/2264924160*x^12*y^7 - 164021/1132462080*x^10*y^7 + 12617/301989888*x^8*y^7 - 2387/377487360*x^6*y^7 + 341/754974720*x^4*y^7 - 31/2642411520*x^2*y^7 + 1/21139292160*y^7
23
8:
39738887544947716125/2048*x^15*y^8 - 92724070938211337625/2048*x^13*y^8 + 84098575967214934125/2048*x^11*y^8 - 37605054294283100625/2048*x^9*y^8 + 8678089452526869375/2048*x^7*y^8 - 985080424340887875/2048*x^5*y^8 + 46908591635280375/2048*x^3*y^8 - 609202488769875/2048*x*y^8
23
-8:
65231/21139292160*x^15*y^8 - 65231/9059696640*x^13*y^8 + 19721/3019898880*x^11*y^8 - 5291/1811939328*x^9*y^8 + 407/603979776*x^7*y^8 - 77/1006632960*x^5*y^8 + 11/3019898880*x^3*y^8 - 1/21139292160*x*y^8
23
9:
-596083313174215741875/2048*x^14*y^9 + 1205412922196747389125/2048*x^12*y^9 - 925084335639364275375/2048*x^10*y^9 + 338445488648547905625/2048*x^8*y^9 - 60746626167688085625/2048*x^6*y^9 + 4925402121704439375/2048*x^4*y^9 - 140725774905841125/2048*x^2*y^9 + 609202488769875/2048*y^9
23
-9:
65231/676457349120*x^14*y^9 - 848003/4348654387200*x^12*y^9 + 216931/1449551462400*x^10*y^9 - 5291/96636764160*x^8*y^9 + 2849/289910292480*x^6*y^9 - 77/96636764160*x^4*y^9 + 11/483183820800*x^2*y^9 - 1/10146860236800*y^9
23
10:
4172583192219510193125/1024*x^13*y^10 - 3616238766590242167375/512*x^11*y^10 + 4625421678196821376875/1024*x^9*y^10 - 338445488648547905625/256*x^7*y^10 + 182239878503064256875/1024*x^5*y^10 - 4925402121704439375/512*x^3*y^10 + 140725774905841125/1024*x*y^10
23
-10:
65231/22323092520960*x^13*y^10 - 848003/167423193907200*x^11*y^10 + 19721/6088116142080*x^9*y^10 - 481/507343011840*x^7*y^10 + 37/289910292480*x^5*y^10 - 1/144955146240*x^3*y^10 + 1/10146860236800*x*y^10
23
11:
-54243581498853632510625/1024*x^12*y^11 + 39778626432492663841125/512*x^10*y^11 - 41628795103771392391875/1024*x^8*y^11 + 2369118420539835339375/256*x^6*y^11 - 911199392515321284375/1024*x^4*y^11 + 14776206365113318125/512*x^2*y^11 - 140725774905841125/1024*y^11
23
-11:
65231/758985145712640*x^12*y^11 - 65231/517489872076800*x^10*y^11 + 1517/22999549870080*x^8*y^11 - 37/2464237486080*x^6*y^11 + 37/25628069855232*x^4*y^11 - 1/21356724879360*x^2*y^11 + 1/4484912224665600*y^11
23
12:
162730744496560897531875/256*x^11*y^12 - 198893132162463319205625/256*x^9*y^12 + 41628795103771392391875/128*x^7*y^12 - 7107355261619506018125/128*x^5*y^12 + 911199392515321284375/256*x^3*y^12 - 14776206365113318125/256*x*y^12
23
-12:
65231/26564480099942400*x^11*y^12 - 65231/21734574627225600*x^9*y^12 + 1517/1207476368179200*x^7*y^12 - 37/172496624025600*x^5*y^12 + 37/2690947334799360*x^3*y^12 - 1/4484912224665600*x*y^12
23
13:
-1790038189462169872850625/256*x^10*y^13 + 1790038189462169872850625/256*x^8*y^13 - 291401565726399746743125/128*x^6*y^13 + 35536776308097530090625/128*x^4*y^13 - 2733598177545963853125/256*x^2*y^13 + 14776206365113318125/256*y^13
23
-13:
65231/956321283597926400*x^10*y^13 - 65231/956321283597926400*x^8*y^13 + 1517/68308663114137600*x^6*y^13 - 37/13661732622827520*x^4*y^13 + 37/355205048193515520*x^2*y^13 - 1/1776025240967577600*y^13
23
14:
8950190947310849364253125/128*x^9*y^14 - 1790038189462169872850625/32*x^7*y^14 + 874204697179199240229375/64*x^5*y^14 - 35536776308097530090625/32*x^3*y^14 + 2733598177545963853125/128*x*y^14
23
-14:
1763/956321283597926400*x^9*y^14 - 1763/1195401604497408000*x^7*y^14 + 41/113847771856896000*x^5*y^14 - 1/34154331557068800*x^3*y^14 + 1/1776025240967577600*x*y^14
23
15:
-80551718525797644278278125/128*x^8*y^15 + 12530267326235189109954375/32*x^6*y^15 - 4371023485895996201146875/64*x^4*y^15 + 106610328924292590271875/32*x^2*y^15 - 2733598177545963853125/128*y^15
23
-15:
1763/36340208776721203200*x^8*y^15 - 1763/58403906962587648000*x^6*y^15 + 41/7787187595011686400*x^4*y^15 - 1/3893593797505843200*x^2*y^15 + 1/607400632410911539200*y^15
23
16:
80551718525797644278278125/16*x^7*y^16 - 37590801978705567329863125/16*x^5*y^16 + 4371023485895996201146875/16*x^3*y^16 - 106610328924292590271875/16*x*y^16
23
-16:
1763/1417268142292126924800*x^7*y^16 - 1763/3037003162054557696000*x^5*y^16 + 41/607400632410911539200*x^3*y^16 - 1/607400632410911539200*x*y^16
23
17:
-563862029680583509947946875/16*x^6*y^17 + 187954009893527836649315625/16*x^4*y^17 - 13113070457687988603440625/16*x^2*y^17 + 106610328924292590271875/16*y^17
23
-17:
1763/56690725691685076992000*x^6*y^17 - 1763/170072177075055230976000*x^4*y^17 + 41/56690725691685076992000*x^2*y^17 - 1/170072177075055230976000*y^17
23
18:
1691586089041750529843840625/8*x^5*y^18 - 187954009893527836649315625/4*x^3*y^18 + 13113070457687988603440625/8*x*y^18
23
-18:
43/56690725691685076992000*x^5*y^18 - 43/255108265612582846464000*x^3*y^18 + 1/170072177075055230976000*x*y^18
23
19:
-8457930445208752649219203125/8*x^4*y^19 + 563862029680583509947946875/4*x^2*y^19 - 13113070457687988603440625/8*y^19
23
-19:
43/2381010479050773233664000*x^4*y^19 - 43/17857578592880799252480000*x^2*y^19 + 1/35715157185761598504960000*y^19
23
20:
8457930445208752649219203125/2*x^3*y^20 - 563862029680583509947946875/2*x*y^20
23
-20:
1/2381010479050773233664000*x^3*y^20 - 1/35715157185761598504960000*x*y^20
23
21:
-25373791335626257947657609375/2*x^2*y^21 + 563862029680583509947946875/2*y^21
23
-21:
1/104764461078234022281216000*x^2*y^21 - 1/4714400748520531002654720000*y^21
23
22:
25373791335626257947657609375*x*y^22
23
-22:
1/4714400748520531002654720000*x*y^22
23
23:
-25373791335626257947657609375*y^23
23
-23:
1/216862434431944426122117120000*y^23
24
0:
8061900920775/4194304*x^24 - 11835556670925/1048576*x^22 + 60755857577415/2097152*x^20 - 44742685812825/1048576*x^18 + 166966608033225/4194304*x^16 - 12843585233325/524288*x^14 + 10529425731825/1048576*x^12 - 1418249180205/524288*x^10 + 1933976154825/4194304*x^8 - 48522699225/1048576*x^6 + 5019589575/2097152*x^4 - 50702925/1048576*x^2 + 676039/4194304
24
1:
-24185702762325/524288*x^23*y + 130191123380175/524288*x^21*y - 303779287887075/524288*x^19*y + 402684172315425/524288*x^17*y - 166966608033225/262144*x^15*y + 89905096633275/262144*x^13*y - 31588277195475/262144*x^11*y + 7091245901025/262144*x^9*y - 1933976154825/524288*x^7*y + 145568097675/524288*x^5*y - 5019589575/524288*x^3*y + 50702925/524288*x*y
24
-1:
322476036831/4194304*x^23*y - 1735881645069/4194304*x^21*y + 4050390505161/4194304*x^19*y - 5369122297539/4194304*x^17*y + 2226221440443/2097152*x^15*y - 1198734621777/2097152*x^13*y + 421177029273/2097152*x^11*y - 94549945347/2097152*x^9*y + 25786348731/4194304*x^7*y - 1940907969/4194304*x^5*y + 66927861/4194304*x^3*y - 676039/4194304*x*y
24
2:
556271163533475/524288*x^22*y^2 - 2734013590983675/524288*x^20*y^2 + 5771806469854425/524288*x^18*y^2 - 6845630929362225/524288*x^16*y^2 + 2504499120498375/262144*x^14*y^2 - 1168766256232575/262144*x^12*y^2 + 347471049150225/262144*x^10*y^2 - 63821213109225/262144*x^8*y^2 + 13537833083775/524288*x^6*y^2 - 727840488375/524288*x^4*y^2 + 15058768725/524288*x^2*y^2 - 50702925/524288*y^2
24
-2:
24805848987/8388608*x^22*y^2 - 121918108851/8388608*x^20*y^2 + 257382674241/8388608*x^18*y^2 - 305267822937/8388608*x^16*y^2 + 111683349855/4194304*x^14*y^2 - 52118896599/4194304*x^12*y^2 + 15494807097/4194304*x^10*y^2 - 2845984977/4194304*x^8*y^2 + 603693783/8388608*x^6*y^2 - 32456655/8388608*x^4*y^2 + 671517/8388608*x^2*y^2 - 2261/8388608*y^2
24
3:
-6118982798868225/262144*x^21*y^3 + 13670067954918375/131072*x^19*y^3 - 51946258228689825/262144*x^17*y^3 + 6845630929362225/32768*x^15*y^3 - 17531493843488625/131072*x^13*y^3 + 3506298768697725/65536*x^11*y^3 - 1737355245751125/131072*x^9*y^3 + 63821213109225/32768*x^7*y^3 - 40613499251325/262144*x^5*y^3 + 727840488375/131072*x^3*y^3 - 15058768725/262144*x*y^3
24
-3:
2756205443/25165824*x^21*y^3 - 2052493415/4194304*x^19*y^3 + 7799474977/8388608*x^17*y^3 - 3083513363/3145728*x^15*y^3 + 2632267505/4194304*x^13*y^3 - 526453501/2097152*x^11*y^3 + 782566015/12582912*x^9*y^3 - 9582441/1048576*x^7*y^3 + 6097917/8388608*x^5*y^3 - 327845/12582912*x^3*y^3 + 2261/8388608*x*y^3
24
4:
128498638776232725/262144*x^20*y^4 - 259731291143449125/131072*x^18*y^4 + 883086389887727025/262144*x^16*y^4 - 102684463940433375/32768*x^14*y^4 + 227909419965352125/131072*x^12*y^4 - 38569286455674975/65536*x^10*y^4 + 15636197211760125/131072*x^8*y^4 - 446748491764575/32768*x^6*y^4 + 203067496256625/262144*x^4*y^4 - 2183521465125/131072*x^2*y^4 + 15058768725/262144*y^4
24
-4:
2756205443/704643072*x^20*y^4 - 5571053555/352321536*x^18*y^4 + 18941582087/704643072*x^16*y^4 - 2202509545/88080384*x^14*y^4 + 698356685/50331648*x^12*y^4 - 118183439/25165824*x^10*y^4 + 15970735/16777216*x^8*y^4 - 3194147/29360128*x^6*y^4 + 1451885/234881024*x^4*y^4 - 46835/352321536*x^2*y^4 + 323/704643072*y^4
24
5:
-642493193881163625/65536*x^19*y^5 + 2337581620291042125/65536*x^17*y^5 - 883086389887727025/16384*x^15*y^5 + 718791247583033625/16384*x^13*y^5 - 683728259896056375/32768*x^11*y^5 + 192846432278374875/32768*x^9*y^5 - 15636197211760125/16384*x^7*y^5 + 1340245475293725/16384*x^5*y^5 - 203067496256625/65536*x^3*y^5 + 2183521465125/65536*x*y^5
24
-5:
95041567/704643072*x^19*y^5 - 115263177/234881024*x^17*y^5 + 653158003/880803840*x^15*y^5 - 15189721/25165824*x^13*y^5 + 4816253/16777216*x^11*y^5 - 4075291/50331648*x^9*y^5 + 110143/8388608*x^7*y^5 - 330429/293601280*x^5*y^5 + 10013/234881024*x^3*y^5 - 323/704643072*x*y^5
24
6:
12207370683742108875/65536*x^18*y^6 - 39738887544947716125/65536*x^16*y^6 + 13246295848315905375/16384*x^14*y^6 - 9344286218579437125/16384*x^12*y^6 + 7521010858856620125/32768*x^10*y^6 - 1735617890505373875/32768*x^8*y^6 + 109453380482320875/16384*x^6*y^6 - 6701227376468625/16384*x^4*y^6 + 609202488769875/65536*x^2*y^6 - 2183521465125/65536*y^6
24
-6:
95041567/21139292160*x^18*y^6 - 34376737/2348810240*x^16*y^6 + 34376737/1761607680*x^14*y^6 - 10392967/754974720*x^12*y^6 + 2788357/503316480*x^10*y^6 - 214489/167772160*x^8*y^6 + 40579/251658240*x^6*y^6 - 5797/587202560*x^4*y^6 + 527/2348810240*x^2*y^6 - 17/21139292160*y^6
24
7:
-109866336153678979875/32768*x^17*y^7 + 39738887544947716125/4096*x^15*y^7 - 92724070938211337625/8192*x^13*y^7 + 28032858655738311375/4096*x^11*y^7 - 37605054294283100625/16384*x^9*y^7 + 1735617890505373875/4096*x^7*y^7 - 328360141446962625/8192*x^5*y^7 + 6701227376468625/4096*x^3*y^7 - 609202488769875/32768*x*y^7
24
-7:
3065857/21139292160*x^17*y^7 - 1108927/2642411520*x^15*y^7 + 1108927/2264924160*x^13*y^7 - 335257/1132462080*x^11*y^7 + 89947/905969664*x^9*y^7 - 6919/377487360*x^7*y^7 + 1309/754974720*x^5*y^7 - 187/2642411520*x^3*y^7 + 17/21139292160*x*y^7
24
8:
1867727714612542657875/32768*x^16*y^8 - 596083313174215741875/4096*x^14*y^8 + 1205412922196747389125/8192*x^12*y^8 - 308361445213121425125/4096*x^10*y^8 + 338445488648547905625/16384*x^8*y^8 - 12149325233537617125/4096*x^6*y^8 + 1641800707234813125/8192*x^4*y^8 - 20103682129405875/4096*x^2*y^8 + 609202488769875/32768*y^8
24
-8:
3065857/676457349120*x^16*y^8 - 65231/5637144576*x^14*y^8 + 848003/72477573120*x^12*y^8 - 216931/36238786560*x^10*y^8 + 5291/3221225472*x^8*y^8 - 2849/12079595520*x^6*y^8 + 77/4831838208*x^4*y^8 - 11/28185722880*x^2*y^8 + 1/676457349120*y^8
24
9:
-1867727714612542657875/2048*x^15*y^9 + 4172583192219510193125/2048*x^13*y^9 - 3616238766590242167375/2048*x^11*y^9 + 1541807226065607125625/2048*x^9*y^9 - 338445488648547905625/2048*x^7*y^9 + 36447975700612851375/2048*x^5*y^9 - 1641800707234813125/2048*x^3*y^9 + 20103682129405875/2048*x*y^9
24
-9:
3065857/22323092520960*x^15*y^9 - 65231/212600881152*x^13*y^9 + 848003/3189013217280*x^11*y^9 - 19721/173946175488*x^9*y^9 + 481/19327352832*x^7*y^9 - 259/96636764160*x^5*y^9 + 7/57982058496*x^3*y^9 - 1/676457349120*x*y^9
24
10:
28015915719188139868125/2048*x^14*y^10 - 54243581498853632510625/2048*x^12*y^10 + 39778626432492663841125/2048*x^10*y^10 - 13876265034590464130625/2048*x^8*y^10 + 2369118420539835339375/2048*x^6*y^10 - 182239878503064256875/2048*x^4*y^10 + 4925402121704439375/2048*x^2*y^10 - 20103682129405875/2048*y^10
24
-10:
3065857/758985145712640*x^14*y^10 - 848003/108426449387520*x^12*y^10 + 848003/147854249164800*x^10*y^10 - 19721/9856949944320*x^8*y^10 + 3367/9856949944320*x^6*y^10 - 259/9856949944320*x^4*y^10 + 7/9856949944320*x^2*y^10 - 1/344993248051200*y^10
24
11:
-196111410034316979076875/1024*x^13*y^11 + 162730744496560897531875/512*x^11*y^11 - 198893132162463319205625/1024*x^9*y^11 + 13876265034590464130625/256*x^7*y^11 - 7107355261619506018125/1024*x^5*y^11 + 182239878503064256875/512*x^3*y^11 - 4925402121704439375/1024*x*y^11
24
-11:
3065857/26564480099942400*x^13*y^11 - 848003/4427413349990400*x^11*y^11 + 848003/7244858209075200*x^9*y^11 - 19721/603738184089600*x^7*y^11 + 481/114997749350400*x^5*y^11 - 37/172496624025600*x^3*y^11 + 1/344993248051200*x*y^11
24
12:
2549448330446120727999375/1024*x^12*y^12 - 1790038189462169872850625/512*x^10*y^12 + 1790038189462169872850625/1024*x^8*y^12 - 97133855242133248914375/256*x^6*y^12 + 35536776308097530090625/1024*x^4*y^12 - 546719635509192770625/512*x^2*y^12 + 4925402121704439375/1024*y^12
24
-12:
3065857/956321283597926400*x^12*y^12 - 65231/14489716418150400*x^10*y^12 + 65231/28979432836300800*x^8*y^12 - 1517/3104939232460800*x^6*y^12 + 37/827983795322880*x^4*y^12 - 37/26909473347993600*x^2*y^12 + 1/161456840087961600*y^12
24
13:
-7648344991338362183998125/256*x^11*y^13 + 8950190947310849364253125/256*x^9*y^13 - 1790038189462169872850625/128*x^7*y^13 + 291401565726399746743125/128*x^5*y^13 - 35536776308097530090625/256*x^3*y^13 + 546719635509192770625/256*x*y^13
24
-13:
82861/956321283597926400*x^11*y^13 - 1763/17387659701780480*x^9*y^13 + 1763/43469149254451200*x^7*y^13 - 41/6209878464921600*x^5*y^13 + 1/2483951385968640*x^3*y^13 - 1/161456840087961600*x*y^13
24
14:
84131794904721984023979375/256*x^10*y^14 - 80551718525797644278278125/256*x^8*y^14 + 12530267326235189109954375/128*x^6*y^14 - 1457007828631998733715625/128*x^4*y^14 + 106610328924292590271875/256*x^2*y^14 - 546719635509192770625/256*y^14
24
-14:
82861/36340208776721203200*x^10*y^14 - 1763/807560195038248960*x^8*y^14 + 1763/2595729198337228800*x^6*y^14 - 41/519145839667445760*x^4*y^14 + 1/346097226444963840*x^2*y^14 - 1/67488959156767948800*y^14
24
15:
-420658974523609920119896875/128*x^9*y^15 + 80551718525797644278278125/32*x^7*y^15 - 37590801978705567329863125/64*x^5*y^15 + 1457007828631998733715625/32*x^3*y^15 - 106610328924292590271875/128*x*y^15
24
-15:
82861/1417268142292126924800*x^9*y^15 - 1763/39368559508114636800*x^7*y^15 + 1763/168722397891919872000*x^5*y^15 - 41/50616719367575961600*x^3*y^15 + 1/67488959156767948800*x*y^15
24
16:
3785930770712489281079071875/128*x^8*y^16 - 563862029680583509947946875/32*x^6*y^16 + 187954009893527836649315625/64*x^4*y^16 - 4371023485895996201146875/32*x^2*y^16 + 106610328924292590271875/128*y^16
24
-16:
82861/56690725691685076992000*x^8*y^16 - 1763/2024668774703038464000*x^6*y^16 + 1763/12148012648218230784000*x^4*y^16 - 41/6074006324109115392000*x^2*y^16 + 1/24296025296436461568000*y^16
24
17:
-3785930770712489281079071875/16*x^7*y^17 + 1691586089041750529843840625/16*x^5*y^17 - 187954009893527836649315625/16*x^3*y^17 + 4371023485895996201146875/16*x*y^17
24
-17:
2021/56690725691685076992000*x^7*y^17 - 43/2699558366270717952000*x^5*y^17 + 43/24296025296436461568000*x^3*y^17 - 1/24296025296436461568000*x*y^17
24
18:
26501515394987424967553503125/16*x^6*y^18 - 8457930445208752649219203125/16*x^4*y^18 + 563862029680583509947946875/16*x^2*y^18 - 4371023485895996201146875/16*y^18
24
-18:
2021/2381010479050773233664000*x^6*y^18 - 43/158734031936718215577600*x^4*y^18 + 43/2381010479050773233664000*x^2*y^18 - 1/7143031437152319700992000*y^18
24
19:
-79504546184962274902660509375/8*x^5*y^19 + 8457930445208752649219203125/4*x^3*y^19 - 563862029680583509947946875/8*x*y^19
24
-19:
47/2381010479050773233664000*x^5*y^19 - 1/238101047905077323366400*x^3*y^19 + 1/7143031437152319700992000*x*y^19
24
20:
397522730924811374513302546875/8*x^4*y^20 - 25373791335626257947657609375/4*x^2*y^20 + 563862029680583509947946875/8*y^20
24
-20:
47/104764461078234022281216000*x^4*y^20 - 1/17460743513039003713536000*x^2*y^20 + 1/1571466916173510334218240000*y^20
24
21:
-397522730924811374513302546875/2*x^3*y^21 + 25373791335626257947657609375/2*x*y^21
24
-21:
47/4714400748520531002654720000*x^3*y^21 - 1/1571466916173510334218240000*x*y^21
24
22:
1192568192774434123539907640625/2*x^2*y^22 - 25373791335626257947657609375/2*y^22
24
-22:
47/216862434431944426122117120000*x^2*y^22 - 1/216862434431944426122117120000*y^22
24
23:
-1192568192774434123539907640625*x*y^23
24
-23:
1/216862434431944426122117120000*x*y^23
24
24:
1192568192774434123539907640625*y^24
24
-24:
1/10409396852733332453861621760000*y^24
25
0:
15801325804719/4194304*x^25 - 24185702762325/1048576*x^23 + 130191123380175/2097152*x^21 - 101259762629025/1048576*x^19 + 402684172315425/4194304*x^17 - 33393321606645/524288*x^15 + 29968365544425/1048576*x^13 - 4512611027925/524288*x^11 + 7091245901025/4194304*x^9 - 214886239425/1048576*x^7 + 29113619535/2097152*x^5 - 456326325/1048576*x^3 + 16900975/4194304*x
25
1:
-395033145117975/4194304*x^24*y + 556271163533475/1048576*x^22*y - 2734013590983675/2097152*x^20*y + 1923935489951475/1048576*x^18*y - 6845630929362225/4194304*x^16*y + 500899824099675/524288*x^14*y - 389588752077525/1048576*x^12*y + 49638721307175/524288*x^10*y - 63821213109225/4194304*x^8*y + 1504203675975/1048576*x^6*y - 145568097675/2097152*x^4*y + 1368978975/1048576*x^2*y - 16900975/4194304*y
25
-1:
1215486600363/8388608*x^24*y - 1711603580103/2097152*x^22*y + 8412349510719/4194304*x^20*y - 5919801507543/2097152*x^18*y + 21063479782653/8388608*x^16*y - 1541230227999/1048576*x^14*y + 1198734621777/2097152*x^12*y - 152734527099/1048576*x^10*y + 196372963413/8388608*x^8*y - 4628319003/2097152*x^6*y + 447901839/4194304*x^4*y - 4212243/2097152*x^2*y + 52003/8388608*y
25
2:
1185099435353925/524288*x^23*y^2 - 6118982798868225/524288*x^21*y^2 + 13670067954918375/524288*x^19*y^2 - 17315419409563275/524288*x^17*y^2 + 6845630929362225/262144*x^15*y^2 - 3506298768697725/262144*x^13*y^2 + 1168766256232575/262144*x^11*y^2 - 248193606535875/262144*x^9*y^2 + 63821213109225/524288*x^7*y^2 - 4512611027925/524288*x^5*y^2 + 145568097675/524288*x^3*y^2 - 1368978975/524288*x*y^2
25
-2:
135054066707/25165824*x^23*y^2 - 697319977079/25165824*x^21*y^2 + 519280833995/8388608*x^19*y^2 - 1973267169181/25165824*x^17*y^2 + 780128880839/12582912*x^15*y^2 - 133192735753/4194304*x^13*y^2 + 133192735753/12582912*x^11*y^2 - 28284171685/12582912*x^9*y^2 + 2424357573/8388608*x^7*y^2 - 514257667/25165824*x^5*y^2 + 16588957/25165824*x^3*y^2 - 52003/8388608*x*y^2
25
3:
-27257287013140275/524288*x^22*y^3 + 128498638776232725/524288*x^20*y^3 - 259731291143449125/524288*x^18*y^3 + 294362129962575675/524288*x^16*y^3 - 102684463940433375/262144*x^14*y^3 + 45581883993070425/262144*x^12*y^3 - 12856428818558325/262144*x^10*y^3 + 2233742458822875/262144*x^8*y^3 - 446748491764575/524288*x^6*y^3 + 22563055139625/524288*x^4*y^3 - 436704293025/524288*x^2*y^3 + 1368978975/524288*y^3
25
-3:
19293438101/100663296*x^22*y^3 - 30318259873/33554432*x^20*y^3 + 61281589105/33554432*x^18*y^3 - 208357402957/100663296*x^16*y^3 + 24227604995/16777216*x^14*y^3 - 10754692949/16777216*x^12*y^3 + 9100124803/50331648*x^10*y^3 - 527034255/16777216*x^8*y^3 + 105406851/33554432*x^6*y^3 - 15970735/100663296*x^4*y^3 + 103037/33554432*x^2*y^3 - 323/33554432*y^3
25
4:
299830157144543025/262144*x^21*y^4 - 642493193881163625/131072*x^19*y^4 + 2337581620291042125/262144*x^17*y^4 - 294362129962575675/32768*x^15*y^4 + 718791247583033625/131072*x^13*y^4 - 136745651979211275/65536*x^11*y^4 + 64282144092791625/131072*x^9*y^4 - 2233742458822875/32768*x^7*y^4 + 1340245475293725/262144*x^5*y^4 - 22563055139625/131072*x^3*y^4 + 436704293025/262144*x*y^4
25
-4:
665290969/100663296*x^21*y^4 - 475207835/16777216*x^19*y^4 + 1728947655/33554432*x^17*y^4 - 653158003/12582912*x^15*y^4 + 531640235/16777216*x^13*y^4 - 101141313/8388608*x^11*y^4 + 142635185/50331648*x^9*y^4 - 1652145/4194304*x^7*y^4 + 991287/33554432*x^5*y^4 - 50065/50331648*x^3*y^4 + 323/33554432*x*y^4
25
5:
-6296433300035403525/262144*x^20*y^5 + 12207370683742108875/131072*x^18*y^5 - 39738887544947716125/262144*x^16*y^5 + 4415431949438635125/32768*x^14*y^5 - 9344286218579437125/131072*x^12*y^5 + 1504202171771324025/65536*x^10*y^5 - 578539296835124625/131072*x^8*y^5 + 15636197211760125/32768*x^6*y^5 - 6701227376468625/262144*x^4*y^5 + 67689165418875/131072*x^2*y^5 - 436704293025/262144*y^5
25
-5:
665290969/3019898880*x^20*y^5 - 1805789773/2113929216*x^18*y^5 + 653158003/469762048*x^16*y^5 - 653158003/528482304*x^14*y^5 + 197466373/301989888*x^12*y^5 - 52978783/251658240*x^10*y^5 + 4075291/100663296*x^8*y^5 - 110143/25165824*x^6*y^5 + 110143/469762048*x^4*y^5 - 10013/2113929216*x^2*y^5 + 323/21139292160*y^5
25
6:
31482166500177017625/65536*x^19*y^6 - 109866336153678979875/65536*x^17*y^6 + 39738887544947716125/16384*x^15*y^6 - 30908023646070445875/16384*x^13*y^6 + 28032858655738311375/32768*x^11*y^6 - 7521010858856620125/32768*x^9*y^6 + 578539296835124625/16384*x^7*y^6 - 46908591635280375/16384*x^5*y^6 + 6701227376468625/65536*x^3*y^6 - 67689165418875/65536*x*y^6
25
-6:
21460999/3019898880*x^19*y^6 - 58251283/2348810240*x^17*y^6 + 21069613/587202560*x^15*y^6 - 21069613/754974720*x^13*y^6 + 6369883/503316480*x^11*y^6 - 1708993/503316480*x^9*y^6 + 131461/251658240*x^7*y^6 - 3553/83886080*x^5*y^6 + 3553/2348810240*x^3*y^6 - 323/21139292160*x*y^6
25
7:
-598161163503363334875/65536*x^18*y^7 + 1867727714612542657875/65536*x^16*y^7 - 596083313174215741875/16384*x^14*y^7 + 401804307398915796375/16384*x^12*y^7 - 308361445213121425125/32768*x^10*y^7 + 67689097729709581125/32768*x^8*y^7 - 4049775077845872375/16384*x^6*y^7 + 234542958176401875/16384*x^4*y^7 - 20103682129405875/65536*x^2*y^7 + 67689165418875/65536*y^7
25
-7:
21460999/96636764160*x^18*y^7 - 52119569/75161927680*x^16*y^7 + 3326781/3758096384*x^14*y^7 - 14416051/24159191040*x^12*y^7 + 3687827/16106127360*x^10*y^7 - 269841/5368709120*x^8*y^7 + 48433/8053063680*x^6*y^7 - 187/536870912*x^4*y^7 + 561/75161927680*x^2*y^7 - 17/676457349120*y^7
25
8:
5383450471530270013875/32768*x^17*y^8 - 1867727714612542657875/4096*x^15*y^8 + 4172583192219510193125/8192*x^13*y^8 - 1205412922196747389125/4096*x^11*y^8 + 1541807226065607125625/16384*x^9*y^8 - 67689097729709581125/4096*x^7*y^8 + 12149325233537617125/8192*x^5*y^8 - 234542958176401875/4096*x^3*y^8 + 20103682129405875/32768*x*y^8
25
-8:
21460999/3189013217280*x^17*y^8 - 52119569/2790386565120*x^15*y^8 + 1108927/53150220288*x^13*y^8 - 14416051/1195879956480*x^11*y^8 + 335257/86973087744*x^9*y^8 - 8177/12079595520*x^7*y^8 + 4403/72477573120*x^5*y^8 - 17/7247757312*x^3*y^8 + 17/676457349120*x*y^8
25
9:
-91518658016014590235875/32768*x^16*y^9 + 28015915719188139868125/4096*x^14*y^9 - 54243581498853632510625/8192*x^12*y^9 + 13259542144164221280375/4096*x^10*y^9 - 13876265034590464130625/16384*x^8*y^9 + 473823684107967067875/4096*x^6*y^9 - 60746626167688085625/8192*x^4*y^9 + 703628874529205625/4096*x^2*y^9 - 20103682129405875/32768*y^9
25
-9:
21460999/108426449387520*x^16*y^9 - 3065857/6324876214272*x^14*y^9 + 848003/1807107489792*x^12*y^9 - 848003/3696356229120*x^10*y^9 + 19721/328564998144*x^8*y^9 - 3367/410706247680*x^6*y^9 + 259/492847497216*x^4*y^9 - 1/82141249536*x^2*y^9 + 1/22999549870080*y^9
25
10:
91518658016014590235875/2048*x^15*y^10 - 196111410034316979076875/2048*x^13*y^10 + 162730744496560897531875/2048*x^11*y^10 - 66297710720821106401875/2048*x^9*y^10 + 13876265034590464130625/2048*x^7*y^10 - 1421471052323901203625/2048*x^5*y^10 + 60746626167688085625/2048*x^3*y^10 - 703628874529205625/2048*x*y^10
25
-10:
3065857/542132246937600*x^15*y^10 - 3065857/252995048570880*x^13*y^10 + 848003/84331682856960*x^11*y^10 - 848003/206995948830720*x^9*y^10 + 19721/22999549870080*x^7*y^10 - 481/5476083302400*x^5*y^10 + 37/9856949944320*x^3*y^10 - 1/22999549870080*x*y^10
25
11:
-1372779870240218853538125/2048*x^14*y^11 + 2549448330446120727999375/2048*x^12*y^11 - 1790038189462169872850625/2048*x^10*y^11 + 596679396487389957616875/2048*x^8*y^11 - 97133855242133248914375/2048*x^6*y^11 + 7107355261619506018125/2048*x^4*y^11 - 182239878503064256875/2048*x^2*y^11 + 703628874529205625/2048*y^11
25
-11:
3065857/19516760889753600*x^14*y^11 - 39856141/136617326228275200*x^12*y^11 + 848003/4139918976614400*x^10*y^11 - 848003/12419756929843200*x^8*y^11 + 19721/1774250989977600*x^6*y^11 - 481/591416996659200*x^4*y^11 + 37/1774250989977600*x^2*y^11 - 1/12419756929843200*y^11
25
12:
9609459091681531974766875/1024*x^13*y^12 - 7648344991338362183998125/512*x^11*y^12 + 8950190947310849364253125/1024*x^9*y^12 - 596679396487389957616875/256*x^7*y^12 + 291401565726399746743125/1024*x^5*y^12 - 7107355261619506018125/512*x^3*y^12 + 182239878503064256875/1024*x*y^12
25
-12:
82861/19516760889753600*x^13*y^12 - 1077193/159386880599654400*x^11*y^12 + 22919/5795886567260160*x^9*y^12 - 22919/21734574627225600*x^7*y^12 + 533/4139918976614400*x^5*y^12 - 13/2069959488307200*x^3*y^12 + 1/12419756929843200*x*y^12
25
13:
-124922968191859915671969375/1024*x^12*y^13 + 84131794904721984023979375/512*x^10*y^13 - 80551718525797644278278125/1024*x^8*y^13 + 4176755775411729703318125/256*x^6*y^13 - 1457007828631998733715625/1024*x^4*y^13 + 21322065784858518054375/512*x^2*y^13 - 182239878503064256875/1024*y^13
25
-13:
82861/741636913810636800*x^12*y^13 - 82861/550609223889715200*x^10*y^13 + 1763/24471521061765120*x^8*y^13 - 1763/117987690833510400*x^6*y^13 + 41/31463384222269440*x^4*y^13 - 1/26219486851891200*x^2*y^13 + 1/6135359923342540800*y^13
25
14:
374768904575579747015908125/256*x^11*y^14 - 420658974523609920119896875/256*x^9*y^14 + 80551718525797644278278125/128*x^7*y^14 - 12530267326235189109954375/128*x^5*y^14 + 1457007828631998733715625/256*x^3*y^14 - 21322065784858518054375/256*x*y^14
25
-14:
82861/28923839638614835200*x^11*y^14 - 82861/25768511678038671360*x^9*y^14 + 1763/1431583982113259520*x^7*y^14 - 1763/9203039885013811200*x^5*y^14 + 41/3681215954005524480*x^3*y^14 - 1/6135359923342540800*x*y^14
25
15:
-4122457950331377217174989375/256*x^10*y^15 + 3785930770712489281079071875/256*x^8*y^15 - 563862029680583509947946875/128*x^6*y^15 + 62651336631175945549771875/128*x^4*y^15 - 4371023485895996201146875/256*x^2*y^15 + 21322065784858518054375/256*y^15
25
-15:
82861/1156953585544593408000*x^10*y^15 - 82861/1259793904259668377600*x^8*y^15 + 1763/89985278875690598400*x^6*y^15 - 1763/809867509881215385600*x^4*y^15 + 41/539911673254143590400*x^2*y^15 - 1/2699558366270717952000*y^15
25
16:
20612289751656886085874946875/128*x^9*y^16 - 3785930770712489281079071875/32*x^7*y^16 + 1691586089041750529843840625/64*x^5*y^16 - 62651336631175945549771875/32*x^3*y^16 + 4371023485895996201146875/128*x*y^16
25
-16:
2021/1156953585544593408000*x^9*y^16 - 2021/1574742380324585472000*x^7*y^16 + 43/149975464792817664000*x^5*y^16 - 43/2024668774703038464000*x^3*y^16 + 1/2699558366270717952000*x*y^16
25
17:
-185510607764911974772874521875/128*x^8*y^17 + 26501515394987424967553503125/32*x^6*y^17 - 8457930445208752649219203125/64*x^4*y^17 + 187954009893527836649315625/32*x^2*y^17 - 4371023485895996201146875/128*y^17
25
-17:
2021/48592050592872923136000*x^8*y^17 - 2021/85036088537527615488000*x^6*y^17 + 43/11338145138337015398400*x^4*y^17 - 43/255108265612582846464000*x^2*y^17 + 1/1020433062450331385856000*y^17
25
18:
185510607764911974772874521875/16*x^7*y^18 - 79504546184962274902660509375/16*x^5*y^18 + 8457930445208752649219203125/16*x^3*y^18 - 187954009893527836649315625/16*x*y^18
25
-18:
47/48592050592872923136000*x^7*y^18 - 47/113381451383370153984000*x^5*y^18 + 1/22676290276674030796800*x^3*y^18 - 1/1020433062450331385856000*x*y^18
25
19:
-1298574254354383823410121653125/16*x^6*y^19 + 397522730924811374513302546875/16*x^4*y^19 - 25373791335626257947657609375/16*x^2*y^19 + 187954009893527836649315625/16*y^19
25
-19:
47/2138050226086408617984000*x^6*y^19 - 47/6984297405215601485414400*x^4*y^19 + 1/2328099135071867161804800*x^2*y^19 - 1/314293383234702066843648000*y^19
25
20:
3895722763063151470230364959375/8*x^5*y^20 - 397522730924811374513302546875/4*x^3*y^20 + 25373791335626257947657609375/8*x*y^20
25
-20:
47/96212260173888387809280000*x^5*y^20 - 47/471440074852053100265472000*x^3*y^20 + 1/314293383234702066843648000*x*y^20
25
21:
-19478613815315757351151824796875/8*x^4*y^21 + 1192568192774434123539907640625/4*x^2*y^21 - 25373791335626257947657609375/8*y^21
25
-21:
47/4425763967998865839226880000*x^4*y^21 - 47/36143739071990737687019520000*x^2*y^21 + 1/72287478143981475374039040000*y^21
25
22:
19478613815315757351151824796875/2*x^3*y^22 - 1192568192774434123539907640625/2*x*y^22
25
-22:
1/4425763967998865839226880000*x^3*y^22 - 1/72287478143981475374039040000*x*y^22
25
23:
-58435841445947272053455474390625/2*x^2*y^23 + 1192568192774434123539907640625/2*y^23
25
-23:
1/212436670463945560282890240000*x^2*y^23 - 1/10409396852733332453861621760000*y^23
25
24:
58435841445947272053455474390625*x*y^24
25
-24:
1/10409396852733332453861621760000*x*y^24
25
25:
-58435841445947272053455474390625*y^25
25
-25:
1/520469842636666622693081088000000*y^25
Definition
For $0\leq m\leq\ell$, $P_\ell^m(x)=(-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}P_\ell(x)$, where $P_\ell$ is the ordinary Legendre polynomial; this is the Condon-Shortley convention. Negative orders are given by (3).
Parameters
$\ell$
—   degree ($\ell\geq0$)
$m$
—   order ($|m|\leq\ell$)
Formulas
(1)
$P_\ell^m(x)=(-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}P_\ell(x)$ for $0\leq m\leq\ell$ [1].
(2)
$P_\ell^m(x)=\frac{(-1)^m}{2^\ell\ell!}(1-x^2)^{m/2} \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell$ for $0\leq m\leq\ell$ [1].
(3)
$P_\ell^{-m}(x)=(-1)^m(\ell-m)!/(\ell+m)!\,P_\ell^m(x)$ for $m>0$ [1] [3].
(4)
$(\ell-m+1)P_{\ell+1}^m(x)=(2\ell+1)xP_\ell^m(x) -(\ell+m)P_{\ell-1}^m(x)$ [1].
(5)
For fixed $m\geq0$, $\int_{-1}^1P_\ell^m(x)P_k^m(x)\,dx =2(\ell+m)!/((2\ell+1)(\ell-m)!)$ if $k=\ell$, and is $0$ if $k\ne\ell$ [1].
Comments
(6)
Entries are exact polynomials in $x$ and $y$, where $y=(1-x^2)^{1/2}$. Substituting this value of $y$ gives the usual function $P_\ell^m(x)$ on $-1\leq x\leq1$.
(7)
These entries carry the Condon-Shortley phase $(-1)^m$, so they are DLMF's Ferrers function of the first kind $\mathsf{P}_\ell^m$ (DLMF 14.7.8) [2]. DLMF writes $P_\ell^m$ for a different function, $P_\ell^m(x)=(x^2-1)^{m/2}\frac{d^m}{dx^m}P_\ell(x)$ (DLMF 14.7.11), with no Condon-Shortley phase.
Programs
(P1)
Sage
gen_legendre_P(26, 1, x)   # the next row after this table
Links
Similar tables
Legendre polynomials —   the rows with $m=0$ are the ordinary Legendre polynomials
Gegenbauer polynomials —   for $m\geq0$, $P_\ell^m(x)=(-1)^m(2m)!/(2^m m!)\,(1-x^2)^{m/2}\,C_{\ell-m}^{m+1/2}(x)$
Data properties
Entries are of type: rational polynomial
How well the digits are known: exact
Table is complete: no (it holds every row with $0\leq\ell\leq25$ and $|m|\leq\ell$; the range is set by the size of the entries, since $\ell=25$ already gives 676 rows and a 115 KB entries block)
Repeats values from: the Legendre polynomials
How they were obtained:

Every entry is exact, a polynomial over $\mathbb{Q}$ in $x$ and $y$ with $y=(1-x^2)^{1/2}$. The generator computes the ordinary Legendre polynomials by Bonnet's recurrence, applies (1) for nonnegative order, and applies (3) for negative order.

more

Before entries are written, it checks the initial rows through $\ell=3$, the negative-order relation, the recurrence (4), the exact orthogonality relation (5), and the $m=0$ rows against the generated Legendre polynomials. The build also read the published Legendre polynomial table and checked that every stored row with $m=0$ agrees with it through $\ell=25$.



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