%I #41 Feb 16 2025 08:33:20
%S 1,4,4,0,6,16,8,0,17,40,28,0,38,96,56,0,84,204,124,0,172,400,232,0,
%T 325,760,448,0,594,1376,784,0,1049,2404,1388,0,1796,4096,2320,0,3005,
%U 6808,3864,0,4912,11072,6216,0,7877,17688,9940,0,12430,27792,15488,0
%N Expansion of (phi(x) / f(-x^4))^2 in powers of x where phi(), f() 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="/A227033/b227033.txt">Table of n, a(n) for n = 0..2500</a> (terms 0..55 from Michael Somos)
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(1/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
%F Euler transform of period 4 sequence [4, -6, 4, 0, ...].
%F a(4*n + 3) = 0. a(2*n) = A112160(n). a(4*n + 1) = 4 * A022569(n).
%e G.f. = 1 + 4*x + 4*x^2 + 6*x^4 + 16*x^5 + 8*x^6 + 17*x^8 + 40*x^9 + 28*x^10 + ...
%e G.f. = 1/q + 4*q^2 + 4*q^5 + 6*q^11 + 16*q^14 + 8*q^17 + 17*q^23 + 40*q^26 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / QPochhammer[ x^4])^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))};
%Y Cf. A022569, A112160.
%K nonn,changed
%O 0,2
%A _Michael Somos_, Jul 03 2013