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%I #28 Mar 04 2018 13:36:42
%S 1,6,738,402444,138030342,63625789080,27583809566796,
%T 12841110779519280,5988752245273028886,2859827345620916000346,
%U 1377856546809576262931880,671500179383482897207038108,329754232921005442388958831684
%N Coefficients in expansion of (E_4^3/E_6^2)^(1/288).
%C In general, for m > 0, the expansion of (E_4^3/E_6^2)^m is asymptotic to 2^(8*m) * Pi^(6*m) * exp(2*Pi*n) / (3^m * Gamma(1/4)^(8*m) * Gamma(2*m) * n^(1-2*m)). - _Vaclav Kotesovec_, Mar 04 2018
%H Seiichi Manyama, <a href="/A289365/b289365.txt">Table of n, a(n) for n = 0..368</a>
%F G.f.: Product_{n>=1} (1-q^n)^(-A289367(n)).
%F a(n) ~ c * exp(2*Pi*n) / n^(143/144), where c = 2^(1/36) * Pi^(1/48) / (3^(1/288) * Gamma(1/144) * Gamma(1/4)^(1/36)) = 0.00699657322237604876174085217217686... - _Vaclav Kotesovec_, Jul 08 2017, updated Mar 04 2018
%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Feb 25 2018
%F a(n) * A289366(n) ~ -sin(Pi/144) * exp(4*Pi*n) / (144*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t nmax = 20; CoefficientList[Series[((1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2)^(1/288), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 08 2017 *)
%Y (E_4^3/E_6^2)^(k/288): this sequence (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), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
%Y Cf. A289209 (E_4^3/E_6^2), A289366, A289367, A300025.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 04 2017
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