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A245669 Expansion of q * f(q, q^5)^3 in powers of q where f() is Ramanujan's two-variable theta function.

%I A245669 #13 Feb 16 2025 08:33:23
%S A245669 1,3,3,1,0,3,6,3,3,6,6,3,0,6,6,1,6,9,6,0,0,12,12,3,7,6,9,6,0,12,6,3,6,
%T A245669 6,12,3,0,12,12,6,6,12,18,6,0,12,12,3,7,15,12,0,0,9,12,6,12,18,6,6,0,
%U A245669 18,18,1,12,12,18,6,0,12,12,9,12,18,15,6,0,18
%N A245669 Expansion of q * f(q, q^5)^3 in powers of q where f() is Ramanujan's two-variable theta function.
%C A245669 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A245669 G. C. Greubel, <a href="/A245669/b245669.txt">Table of n, a(n) for n = 1..1000</a>
%H A245669 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A245669 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A245669 Expansion of q * (chi(q) * psi(-q^3))^3 in powers of q where chi(), psi() are Ramanujan theta functions.
%F A245669 Expansion of (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^3 in powers of q.
%F A245669 Euler transform of period 12 sequence [ 3, -3, 0, 0, 3, -3, 3, 0, 0, -3, 3, -3, ...].
%F A245669 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A245668.
%F A245669 G.f.: x * (Sum_{k in Z} x^(3*k^2 + 2*k))^3.
%F A245669 Convolution cube of A089801.
%F A245669 a(2*n + 2) = A005885(n).
%e A245669 G.f. = q + 3*q^2 + 3*q^3 + q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + 6*q^10 + ...
%t A245669 a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 3, 0, q^(1/3)] - EllipticTheta[ 3, 0, q^3]) / 2)^3, {q, 0, n}];
%t A245669 a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)])^3 / (2^(3/2) q^(1/8)), {q, 0, n}];
%o A245669 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^3, n))};
%Y A245669 Cf. A005885, A089801, A245668.
%K A245669 nonn
%O A245669 1,2
%A A245669 _Michael Somos_, Jul 28 2014