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%I #21 Nov 03 2019 21:04:41
%S 1,-7,41,-253,1555,-9532,58463,-358600,2199546,-13491360,82752059,
%T -507576937,3113328401,-19096245457,117130782240,-718445946527,
%U 4406737223117,-27029636742811,165791883077354,-1016918901125280,6237482995373629,-38258895644996020
%N Convolution inverse of A058537.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Convolution square is A328785. - _Michael Somos_, Nov 02 2019
%H Vaclav Kotesovec, <a href="/A258941/b258941.txt">Table of n, a(n) for n = 0..1260</a>
%F a(n) ~ (-1)^n * c * exp(Pi*n/sqrt(3)), where c = A258942 = 8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / Gamma(1/6)^3 = 1.09786330972731096865822482325074133091288... . - _Vaclav Kotesovec_, Nov 14 2015
%F Expansion of q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3) in powers of q. - _G. C. Greubel_, Jun 22 2018
%F Expansion of q^(-1/6) * 3^(-1/2) * sqrt(b(q)*c(q))/a(q) in powers of q where a(), b(), c() are cubic AGM functions. - _Michael Somos_, Nov 02 2019
%e G.f. = 1 - 7*x + 41*x^2 - 253*x^3 + 1555*x^4 - 9532*x^5 + ... - _Michael Somos_, Nov 02 2019
%e G.f. = q - 7*q^7 + 41*q^13 - 253*q^19 + 1555*q^25 - 9532*q^31 + ... - _Michael Somos_, Nov 02 2019
%t CoefficientList[Series[QPochhammer[x, x] * QPochhammer[x^3, x^3]^2 / (QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3), {x, 0, 50}], x]
%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3), {q,0,60}], q] (* _G. C. Greubel_, Jun 22 2018 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[x] QPochhammer[x^3]^2 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3), {x, 0, n}]; (* _Michael Somos_, Nov 02 2019 *)
%o (PARI) q='q+O('q^50); A = eta(q)*eta(q^3)^2/(eta(q)^3 + 9*q*eta(q^9)^3); Vec(A) \\ _G. C. Greubel_, Jun 22 2018
%Y Cf. A058537, A051273, A058092, A115784, A258942, A328785.
%K sign
%O 0,2
%A _Vaclav Kotesovec_, Nov 07 2015
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