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%I #9 Mar 12 2021 22:24:47
%S 1,-4,12,-32,66,-128,232,-384,639,-1024,1596,-2496,3774,-5632,8328,
%T -12032,17283,-24576,34520,-48288,66882,-91904,125568,-170112,229244,
%U -307200,409236,-542912,716412,-941056,1231048,-1602816,2079237,-2686976,3459264,-4439616
%N Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A233458/b233458.txt">Table of n, a(n) for n = -1..2500</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1) * (phi(-q^4)^2 / (phi(q) * psi(q^4)))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%F Expansion of (eta(q)^2 * eta(q^4)^7 / (eta(q^2)^5 * eta(q^8)^4))^2 in powers of q.
%F Euler transform of period 8 sequence [ -4, 6, -4, -8, -4, 6, -4, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A232772.
%F a(2*n) = -4 * A014969(n). a(2*n - 1) = A112142(n).
%e G.f. = 1/q - 4 + 12*q - 32*q^2 + 66*q^3 - 128*q^4 + 232*q^5 - 384*q^6 + ...
%t a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q^4]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]))^2, {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 2, Pi/4, q]^2)^2, {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^5 * eta(x^8 + A)^4))^2, n))}
%Y Cf. A014969, A112142, A232772.
%K sign
%O -1,2
%A _Michael Somos_, Dec 10 2013
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