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A227033 Expansion of (phi(x) / f(-x^4))^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A227033 #41 Feb 16 2025 08:33:20
%S A227033 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 A227033 325,760,448,0,594,1376,784,0,1049,2404,1388,0,1796,4096,2320,0,3005,
%U A227033 6808,3864,0,4912,11072,6216,0,7877,17688,9940,0,12430,27792,15488,0
%N A227033 Expansion of (phi(x) / f(-x^4))^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A227033 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A227033 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 A227033 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A227033 Expansion of q^(1/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
%F A227033 Euler transform of period 4 sequence [4, -6, 4, 0, ...].
%F A227033 a(4*n + 3) = 0. a(2*n) = A112160(n). a(4*n + 1) = 4 * A022569(n).
%e A227033 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 A227033 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 A227033 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / QPochhammer[ x^4])^2, {x, 0, n}];
%o A227033 (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 A227033 Cf. A022569, A112160.
%K A227033 nonn
%O A227033 0,2
%A A227033 _Michael Somos_, Jul 03 2013