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A263189 Coefficient of y^0 in G(x,y)^4 where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n. 1

%I

%S 1,-12,18,20,24,-96,-4,-120,138,-116,456,-132,356,-540,900,-1884,1440,

%T -2076,1546,-3204,5772,-6572,7860,-7440,12240,-13128,15828,-18632,

%U 23916,-31344,34008,-45084,44118,-44940,53748,-95316,125500,-119136,103632,-100772,156048,-238668,332896,-334596,293616,-253552,335352,-591780,833340,-789012,572634,-678492,998508,-1350228,1632840,-1863108,1749036,-1538120,1841244,-2950512,3959160,-3671160,3113532,-3879628,5512488,-6007836,6159684,-7625868,9420576,-8502888,7725780,-12080952,16859826,-14906736,12391572,-18666168,26180532,-23512824,18486132,-29125692,43754556

%N Coefficient of y^0 in G(x,y)^4 where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n.

%C Compare to the coefficient of y^0 in G(x,y)^2, which equals theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 +...+ 2*(-x)^(n^2) +..., where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n.

%C Note the curious identity: Sum_{n=-oo..+oo} (1-x^n)^n * x^n = 0.

%e G.f.: A(x) = 1 - 12*x + 18*x^2 + 20*x^3 + 24*x^4 - 96*x^5 - 4*x^6 - 120*x^7 + 138*x^8 - 116*x^9 + 456*x^10 - 132*x^11 + 356*x^12 - 540*x^13 + 900*x^14 +...

%e such that A(x) is the coefficient of y^0 in G(x,y)^4 where

%e G(x,y) = N(x,y) + P(x,y), with

%e P(x,y) = 1 + x*y*(1-x) + (x*y)^2*(1-x^2)^2 + (x*y)^3*(1-x^3)^3 + (x*y)^4*(1-x^4)^4 + (x*y)^5*(1-x^5)^5 + (x*y)^6*(1-x^6)^6 +...+ (x*y)^n*(1-x^n)^n +...

%e N(x,y) = (-1/y)/(1-x) + (x/y)^2/(1-x^2)^2 + (-x^2/y)^3/(1-x^3)^3 + (x^3/y)^4/(1-x^4)^4 + (-x^4/y)^5/(1-x^5)^5 +...+ (-x^(n-1)/y)^n/(1-x^n)^n +...

%o (PARI) {a(n) = my(A=sum(m=-sqrtint(n)-1,n+1, x^m*(1-x^m)^m*y^m +x*O(x^n))); polcoeff(polcoeff(A^4,0,y),n,x)}

%o for(n=0,80,print1(a(n),", "))

%Y Cf. A263188.

%K sign

%O 0,2

%A _Paul D. Hanna_, Nov 05 2015

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Last modified June 24 20:02 EDT 2021. Contains 345425 sequences. (Running on oeis4.)