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%I #13 Mar 12 2021 22:24:48
%S 1,1,-2,-4,-3,4,12,8,-10,-28,-18,24,60,38,-48,-120,-75,92,228,140,
%T -172,-416,-252,304,732,439,-524,-1252,-744,884,2088,1232,-1450,-3408,
%U -1998,2336,5460,3182,-3704,-8600,-4986,5772,13344,7700,-8872,-20424,-11736
%N Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in powers of x where f(,) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A261326/b261326.txt">Table of n, a(n) for n = 0..1000</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 f(x) * f(-x^2) * f(-x^6) / f(x^3)^3 in powers of x where f() is a Ramanujan theta function.
%F Euler transform of period 12 sequence [ 1, -3, -2, -2, 1, 2, 1, -2, -2, -3, 1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261325.
%F a(3*n) = A261320(n). a(3*n + 1) = A261325(n).
%e G.f. = 1 + x - 2*x^2 - 4*x^3 - 3*x^4 + 4*x^5 + 12*x^6 + 8*x^7 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ -x^3]^3, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^8), n))};
%Y Cf. A261320, A261325.
%K sign
%O 0,3
%A _Michael Somos_, Aug 14 2015
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