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A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.

%I A254372 #13 Feb 16 2025 08:33:24
%S A254372 1,2,0,-2,1,6,0,-10,3,20,0,-30,1,52,0,-78,6,126,0,-184,3,280,0,-402,
%T A254372 12,590,0,-830,5,1182,0,-1636,21,2280,0,-3108,10,4252,0,-5722,36,7710,
%U A254372 0,-10252,15,13632,0,-17940,60,23586,0,-30744,26,40014,0,-51714,96
%N A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.
%C A254372 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A254372 G. C. Greubel, <a href="/A254372/b254372.txt">Table of n, a(n) for n = 0..1000</a>
%H A254372 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A254372 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A254372 Expansion of eta(q^2)^5 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^5 * eta(q^6)) in powers of q.
%F A254372 Euler transform of period 12 sequence [ 2, -3, 0, 2, 2, -4, 2, 2, 0, -3, 2, 0, ...].
%F A254372 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (4/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A230256.
%F A254372 a(4*n + 2) = 0. a(2*n + 1) = 2 * A254346(n). a(4*n) = A132180(n).
%e A254372 G.f. = 1 + 2*q - 2*q^3 + q^4 + 6*q^5 - 10*q^7 + 3*q^8 + 20*q^9 + ...
%t A254372 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] * EllipticTheta[ 4, 0, q^3] QPochhammer[ q^12] / QPochhammer[ q^4]^3, {q, 0, n}];
%o A254372 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};
%Y A254372 Cf. A132180, A254346, A230256.
%K A254372 sign
%O A254372 0,2
%A A254372 _Michael Somos_, Jan 29 2015