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A263502 Expansion of phi(q) * f(-q^2)^3 / f(-q^6) in powers of q where phi(), f() are Ramanujan theta functions.

%I A263502 #12 Feb 16 2025 08:33:27
%S A263502 1,2,-3,-6,2,0,0,12,-3,-4,12,-6,-6,0,-6,0,2,-6,-12,12,0,0,24,-12,0,14,
%T A263502 -6,-6,12,0,-24,12,-3,0,12,-12,-4,0,-12,-24,12,-6,0,36,-6,0,24,-12,-6,
%U A263502 14,-15,0,0,0,0,24,-6,-24,36,-6,0,0,-18,-24,2,-12,-24,36
%N A263502 Expansion of phi(q) * f(-q^2)^3 / f(-q^6) in powers of q where phi(), f() are Ramanujan theta functions.
%C A263502 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A263502 G.f. is a period 1 Fourier series which satisfies f(-1 / (36* t)) = 17496^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261444.
%H A263502 G. C. Greubel, <a href="/A263502/b263502.txt">Table of n, a(n) for n = 0..1000</a>
%H A263502 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263502 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263502 Expansion of eta(q^2)^8 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
%F A263502 Euler transform of period 12 sequence [2, -6, 2, -4, 2, -5, 2, -4, 2, -6, 2, -3, ...].
%F A263502 a(n) = A263456(4*n). a(8*n + 5) = a(9*n + 6) = 0.
%F A263502 a(3*n + 2) = -3 * A261444(n).
%e A263502 G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 + 12*x^7 - 3*x^8 - 4*x^9 + ...
%t A263502 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] QPochhammer[ q^2]^3 / QPochhammer[ q^6], {q, 0, n}];
%o A263502 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)), n))};
%Y A263502 Cf. A261444, A263456.
%K A263502 sign
%O A263502 0,2
%A A263502 _Michael Somos_, Oct 19 2015