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A262968 Expansion of phi(-q^6) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

%I A262968 #12 Feb 16 2025 08:33:27
%S A262968 1,2,4,8,14,24,38,60,92,138,204,296,424,600,840,1164,1598,2176,2940,
%T A262968 3944,5256,6960,9164,12000,15634,20270,26160,33616,43020,54840,69648,
%U A262968 88140,111164,139748,175136,218832,272646,338760,419792,518880,639780,786976,965820
%N A262968 Expansion of phi(-q^6) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
%C A262968 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262968 G. C. Greubel, <a href="/A262968/b262968.txt">Table of n, a(n) for n = 0..1000</a>
%H A262968 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262968 Expansion of eta(q^2) * eta(q^6)^2 / (eta(q)^2 * eta(q^12)) in powers of q.
%F A262968 Euler transform of period 12 sequence [ 2, 1, 2, 1, 2, -1, 2, 1, 2, 1, 2, 0, ...].
%F A262968 a(n) = A262967(3*n).
%F A262968 a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(9/4) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2015
%e A262968 G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 38*q^6 + 60*q^7 + ...
%t A262968 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 4, 0, q], {q, 0, n}];
%o A262968 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^12 + A)), n))};
%Y A262968 Cf. A262967.
%K A262968 nonn
%O A262968 0,2
%A A262968 _Michael Somos_, Oct 05 2015