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A299993 Coefficients in expansion of (E_4^3/E_6^2)^(1/9).

%I A299993 #18 Mar 04 2018 12:38:13
%S A299993 1,192,41472,18342144,7524397056,3440911653504,1589472997005312,
%T A299993 756816895536990720,364982499184388898816,178417371665487543380928,
%U A299993 88017286719942539086814208,43770603489875525093472688896,21905830503405563891572154843136
%N A299993 Coefficients in expansion of (E_4^3/E_6^2)^(1/9).
%H A299993 Seiichi Manyama, <a href="/A299993/b299993.txt">Table of n, a(n) for n = 0..367</a>
%F A299993 Convolution inverse of A299863.
%F A299993 a(n) ~ 2^(8/9) * Pi^(2/3) * exp(2*Pi*n) / (3^(1/9) * Gamma(2/9) * Gamma(1/4)^(8/9) * n^(7/9)). - _Vaclav Kotesovec_, Mar 04 2018
%F A299993 a(n) * A299863(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A299993 terms = 13;
%t A299993 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A299993 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A299993 (E4[x]^3/E6[x]^2)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)
%Y A299993 (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), this sequence (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
%Y A299993 Cf. A004009 (E_4), A013973 (E_6), A299863.
%K A299993 nonn
%O A299993 0,2
%A A299993 _Seiichi Manyama_, Feb 22 2018