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A299863 Coefficients in expansion of (E_6^2/E_4^3)^(1/9).

%I A299863 #17 Mar 04 2018 12:43:51
%S A299863 1,-192,-4608,-9494784,-1988603904,-1136127187584,-419383041398784,
%T A299863 -200225564597488128,-88040635024586342400,-41470393697874515307456,
%U A299863 -19381646100387803980004352,-9267227811160245194038205184
%N A299863 Coefficients in expansion of (E_6^2/E_4^3)^(1/9).
%H A299863 Seiichi Manyama, <a href="/A299863/b299863.txt">Table of n, a(n) for n = 0..367</a>
%F A299863 G.f.: (1 - 1728/j)^(1/9), where j is the j-function.
%F A299863 a(n) ~ -2^(1/9) * Gamma(1/4)^(8/9) * exp(2*Pi*n) / (3^(17/9) * Pi^(2/3) * Gamma(7/9) * n^(11/9)). - _Vaclav Kotesovec_, Mar 04 2018
%F A299863 a(n) * A299993(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A299863 terms = 12;
%t A299863 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A299863 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A299863 (E6[x]^2/E4[x]^3)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)
%Y A299863 (E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), this sequence (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
%Y A299863 Cf. A000521 (j).
%K A299863 sign
%O A299863 0,2
%A A299863 _Seiichi Manyama_, Feb 21 2018