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A300053 Coefficients in expansion of (E_4^3/E_6^2)^(1/4).

%I A300053 #18 Mar 04 2018 12:37:41
%S A300053 1,432,145152,64494144,29760915456,14274670230432,6975951829890048,
%T A300053 3459591515857458816,1733116511275051696128,875135886353582630388336,
%U A300053 444632598699435462934282752,227042568315636603738176892096
%N A300053 Coefficients in expansion of (E_4^3/E_6^2)^(1/4).
%H A300053 Seiichi Manyama, <a href="/A300053/b300053.txt">Table of n, a(n) for n = 0..366</a>
%F A300053 Convolution inverse of A299861.
%F A300053 a(n) ~ 4 * Pi * exp(2*Pi*n) / (3^(1/4) * Gamma(1/4)^2 * sqrt(n)). - _Vaclav Kotesovec_, Mar 04 2018
%F A300053 a(n) * A299861(n) ~ -exp(4*Pi*n) / (2*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A300053 terms = 12;
%t A300053 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A300053 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A300053 (E4[x]^3/E6[x]^2)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)
%Y A300053 (E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (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), this sequence (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
%Y A300053 Cf. A004009 (E_4), A013973 (E_6), A299861.
%K A300053 nonn
%O A300053 0,2
%A A300053 _Seiichi Manyama_, Feb 23 2018