OFFSET
0,2
COMMENTS
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.
Note the curious identity: Sum_{n=-oo..+oo} (1-x^n)^n * x^n = 0.
EXAMPLE
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 +...
such that A(x) is the coefficient of y^0 in G(x,y)^3 where
G(x,y) = N(x,y) + P(x,y), with
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 +...
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 +...
PROG
(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)}
for(n=0, 80, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 05 2015
STATUS
approved