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%I #14 Mar 12 2021 22:24:45
%S 1,-1,0,-2,1,0,4,-2,0,-6,4,0,10,-6,0,-16,9,0,24,-14,0,-36,20,0,52,-29,
%T 0,-74,42,0,104,-58,0,-144,80,0,198,-110,0,-268,148,0,360,-198,0,-480,
%U 264,0,634,-347,0,-832,454,0,1084,-592,0,-1404,764,0,1808,-982
%N Expansion of psi(-q) / f(q^3) where psi(), 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="/A139136/b139136.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 eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)^3) in powers of q.
%F Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 2, -1, -1, -2, 0, -1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139135.
%F G.f.: Product_{k>0} P(12, x^k) / ( (1 + x^(2*k-1))^2 * P(3, x^k) * P(6, x^k)^2) where P(n, x) is n-th cyclotomic polynomial.
%F a(3*n) = A132002(n). a(3*n + 1) = - A139135(n). a(3*n + 2) = 0.
%F a(n) = (-1)^n * A122792(n). - _Michael Somos_, Sep 07 2015
%e G.f. = 1 - q - 2*q^3 + q^4 + 4*q^6 - 2*q^7 - 6*q^9 + 4*q^10 + 10*q^12 - 6*q^13 + ...
%t a[ n_] := SeriesCoefficient[ 2^(-1/2) q^(-1/8) EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A) * eta(x^6 + A)^3), n))};
%Y Cf. A122792, A132002, A139135.
%K sign
%O 0,4
%A _Michael Somos_, Apr 10 2008
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