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%I #10 Aug 08 2018 22:01:30
%S 1,13,50,72,170,205,362,360,601,650,962,864,1370,1224,1850,1584,2451,
%T 2210,2880,2520,3722,3277,4490,3600,5330,4706,6242,5040,6912,6120,
%U 8500,6624,9410,7813,10610,8424,11882,10250,12672,10440,14521,12506,16130,12240
%N Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A231947/b231947.txt">Table of n, a(n) for n = 0..2500</a>
%F Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3)^2 * eta(q^3) / eta(q) in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231948.
%F -9 * a(n) = A109041(3*n + 1).
%e G.f. = 1 + 13*x + 50*x^2 + 72*x^3 + 170*x^4 + 205*x^5 + 362*x^6 + 360*x^7 + ...
%e G.f. = q + 13*q^4 + 50*q^7 + 72*q^10 + 170*q^13 + 205*q^16 + 362*q^19 + ...
%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)*(eta[q]^3 + 9*eta[q^9]^3)^2*eta[q^3]/eta[q], {q, 0, 50}], q] (* _G. C. Greubel_, Aug 08 2018 *)
%o (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^3 + A) / eta(x + A), n))}
%Y Cf. A109041, A231948.
%K nonn
%O 0,2
%A _Michael Somos_, Nov 15 2013
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