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A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).

%I A299694 #22 Mar 04 2018 12:32:40
%S A299694 1,12,1512,813744,281434656,129501949608,56296822560480,
%T A299694 26218237904433888,12242575532254540032,5850239653863742634172,
%U A299694 2820869122426120317439152,1375631026432164061822527120,675950202173640832786529615232
%N A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).
%H A299694 Seiichi Manyama, <a href="/A299694/b299694.txt">Table of n, a(n) for n = 0..368</a>
%F A299694 Convolution inverse of A296609.
%F A299694 a(n) ~ 2^(1/18) * Pi^(1/24) * exp(2*Pi*n) / (3^(1/144) * Gamma(1/72) * Gamma(1/4)^(1/18) * n^(71/72)). - _Vaclav Kotesovec_, Mar 04 2018
%F A299694 a(n) * A296609(n) ~ -sin(Pi/72) * exp(4*Pi*n) / (72*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A299694 terms = 13;
%t A299694 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A299694 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A299694 (E4[x]^3/E6[x]^2)^(1/144) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%Y A299694 (E_4^3/E_6^2)^(k/288): A289365 (k=1), this sequence (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
%Y A299694 Cf. A004009 (E_4), A013973 (E_6), A296609.
%K A299694 nonn
%O A299694 0,2
%A A299694 _Seiichi Manyama_, Feb 16 2018