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%I #15 Mar 08 2020 13:59:50
%S 1,-3,9,-24,57,-126,264,-528,1017,-1896,3438,-6084,10536,-17898,29880,
%T -49104,79545,-127170,200856,-313692,484830,-742080,1125540,-1692648,
%U 2525160,-3738765,5496246,-8025432,11643576,-16790310,24072048,-34321560,48677625
%N Expansion of b(q^3) * b(q^12) / (b(-q) * b(q^6)) in powers of q where b() is a cubic AGM theta function.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A258111/b258111.txt">Table of n, a(n) for n = 0..1000</a>
%F Expansion of eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18) / (eta(q^2)^9 * eta(q^9) * eta(q^36)) in powers of q.
%F Euler transform of period 36 sequence [ -3, 6, -5, 3, -3, 4, -3, 3, -4, 6, -3, -1, -3, 6, -5, 3, -3, 4, -3, 3, -5, 6, -3, -1, -3, 6, -4, 3, -3, 4, -3, 3, -5, 6, -3, 0, ...]. - Corrected by _Sean A. Irvine_, Mar 06 2020
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A164615.
%F Convolution inverse of A258108.
%F a(n) ~ (-1)^n * exp(4*Pi*sqrt(n)/3) / (2^(5/2) * 3^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e G.f. = 1 - 3*q + 9*q^2 - 24*q^3 + 57*q^4 - 126*q^5 + 264*q^6 - 528*q^7 + ...
%t QP = QPochhammer; A258111[n_] := SeriesCoefficient[(QP[q]^3*QP[q^3]^2 *QP[q^4]^3*QP[q^12]^2*QP[q^18])/(QP[q^2]^9*QP[q^9]*QP[q^36]), {q, 0, n}]; Table[A258111[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 18 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A)), n))};
%Y Cf. A164615, A258108.
%K sign
%O 0,2
%A _Michael Somos_, May 20 2015
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