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%I #7 Nov 05 2015 21:48:58
%S 1,-6,3,6,12,-12,3,-30,18,-48,63,-30,51,-36,129,-168,147,-78,-42,-120,
%T 279,-372,195,-180,339,-294,351,-588,969,-444,90,-840,1617,-888,477,
%U -2082,2487,-1164,669,-2136,2526,-1926,1512,-3402,7233,-5790,765,-5154,12258,-1698,-5754,-8496,14553,-3444,9333,-23886,18234,-6330,4677,-11544,15765,-9456,7059,-23730,42054,-45486,23634,-22452,59061,-16626,-46389,-48780,104931,-6048,45039,-136458,69543,-62280,98535,-29742,48996,-93156
%N Coefficient of y^0 in G(x,y)^3 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 - 6*x + 3*x^2 + 6*x^3 + 12*x^4 - 12*x^5 + 3*x^6 - 30*x^7 + 18*x^8 - 48*x^9 + 63*x^10 - 30*x^11 + 51*x^12 - 36*x^13 + 129*x^14 - 168*x^15 +...
%e such that A(x) is the coefficient of y^0 in G(x,y)^3 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^3,0,y),n,x)}
%o for(n=0,80,print1(a(n),", "))
%Y Cf. A263189.
%K sign
%O 0,2
%A _Paul D. Hanna_, Nov 05 2015
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