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%I #12 Mar 12 2021 22:24:48
%S 1,1,4,5,8,12,17,24,36,48,65,88,116,152,200,257,328,420,532,668,840,
%T 1045,1296,1604,1972,2416,2952,3592,4357,5272,6356,7640,9168,10964,
%U 13080,15576,18497,21920,25932,30604,36048,42392,49752,58288,68184,79617,92820
%N Expansion of f(x^2)^2 / f(-x) in powers of x where f() is a Ramanujan 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="/A260163/b260163.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 q^(-1/8) * eta(q^4)^6 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
%F Euler transform of period 8 sequence [ 1, 3, 1, -3, 1, 3, 1, -1, ...].
%F a(n) = A132965(2*n).
%e G.f. = 1 + x + 4*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 17*x^6 + 24*x^7 + 36*x^8 + ...
%e G.f. = q + q^9 + 4*q^17 + 5*q^25 + 8*q^33 + 12*q^41 + 17*q^49 + 24*q^57 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2]^2 / QPochhammer[ x], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^6 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))};
%Y Cf. A132965.
%K nonn
%O 0,3
%A _Michael Somos_, Nov 09 2015
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