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A231948 Expansion of a(q)^2 * b(q) in powers of q where a(), b() are cubic AGM theta functions.

%I A231948 #9 Aug 08 2018 22:14:47
%S A231948 1,9,0,-90,117,0,-216,450,0,-738,648,0,-1170,1530,0,-1728,1845,0,
%T A231948 -2160,3258,0,-4500,3240,0,-3672,5409,0,-6570,5850,0,-6480,8658,0,
%U A231948 -8640,7776,0,-9594,12330,0,-15300,11016,0,-10800,16650,0,-17280,14256,0,-18450
%N A231948 Expansion of a(q)^2 * b(q) in powers of q where a(), b() are cubic AGM theta functions.
%C A231948 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A231948 G. C. Greubel, <a href="/A231948/b231948.txt">Table of n, a(n) for n = 0..2500</a>
%F A231948 Expansion of (eta(q)^3 + 9 * eta(q^9)^3)^2 * (eta(q) / eta(q^3))^3 in powers of q.
%F A231948 G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(11/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231947.
%F A231948 a(3*n + 2) = 0.  a(3*n + 1) = 9 * A231947(n). 3 * A109041(n) = a(3*n) + A109041(3*n) + A181976(3*n).
%e A231948 G.f. = 1 + 9*q - 90*q^3 + 117*q^4 - 216*q^6 + 450*q^7 - 738*q^9 + ...
%t A231948 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3 + 9*eta[q^9]^3)^2*(eta[q]/eta[q^3])^3, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 08 2018 *)
%o A231948 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3)^2 * (eta(x + A) / eta(x^3 + A))^3, n))}
%Y A231948 Cf. A109041, A181976, A231947.
%K A231948 sign
%O A231948 0,2
%A A231948 _Michael Somos_, Nov 15 2013