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A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.

%I A328785 #12 Nov 08 2019 02:58:59
%S A328785 1,-14,131,-1080,8333,-61580,441893,-3104136,21454675,-146408362,
%T A328785 988876484,-6622729128,44039891747,-291092877360,1914072126008,
%U A328785 -12529113820200,81687913345362,-530724605541430,3437326269162720,-22199991545327616,143016156285625823
%N A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.
%C A328785 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A328785 Convolution square of A258941. Convolution inverse of A058092 with more information there.
%D A328785 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998.
%F A328785 Coefficients of 1/3 of power series in equation (13.23), page 179, [Berndt 1998]
%F A328785 A051273(n) = 3*a(n).
%F A328785 a(n) ~ (-1)^n * 64 * Pi^5 * n * exp(Pi*(3*n+1)/3^(3/2)) / Gamma(1/6)^6. - _Vaclav Kotesovec_, Nov 08 2019
%e A328785 G.f. = 1 - 14*x + 131*x^2 - 1080*x^3 + 8333*x^4 - 61580*x^5 + ...
%e A328785 G.f. = q - 14*q^4 + 131*q^7 - 1080*q^10 + 8333*q^13 - 61580*q^16 + ...
%t A328785 a[ n_] := SeriesCoefficient[ QPochhammer[x]^2 QPochhammer[x^3]^4 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3)^2, {x, 0, n}];
%o A328785 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); A = ((eta(x + A) * eta(x^3 + A)) / (eta(x^2 + A) * eta(x^6 + A)))^2; polcoeff( 1 / (A + x * 16/A^2), n))};
%Y A328785 Cf. A058092, A258941, A051273.
%K A328785 sign
%O A328785 0,2
%A A328785 _Michael Somos_, Nov 02 2019