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A289366 Coefficients in expansion of (E_6^2/E_4^3)^(1/288).

%I A289366 #34 Jun 14 2018 20:27:19
%S A289366 1,-6,-702,-393804,-132734778,-61428055320,-26480146877172,
%T A289366 -12318952616296752,-5730786812846192490,-2732960583228848850522,
%U A289366 -1314627022075990658598360,-639871947654492158944455132,-313833506047227501170833823292
%N A289366 Coefficients in expansion of (E_6^2/E_4^3)^(1/288).
%C A289366 In general, for 0 < m < 1/2, the expansion of (E_6^2/E_4^3)^m is asymptotic to -m * 3^m * Gamma(1/4)^(8*m) * exp(2*n*Pi) / (2^(8*m-1) * Pi^(6*m) * Gamma(1-2*m) * n^(1+2*m)). - _Vaclav Kotesovec_, Mar 04 2018
%H A289366 Seiichi Manyama, <a href="/A289366/b289366.txt">Table of n, a(n) for n = 0..367</a>
%F A289366 G.f.: (1 - 1728/j)^(1/288).
%F A289366 G.f.: Product_{n>=1} (1-q^n)^A289367(n).
%F A289366 a(n) ~ c * exp(2*Pi*n) / n^(145/144), where c = -Gamma(1/4)^(1/36) / (48 * 2^(1/36) * 3^(287/288) * Pi^(1/48) * Gamma(143/144)) = -0.006892157290355982837398273285864980110980721215574657372422958228077... - _Vaclav Kotesovec_, Jul 08 2017, updated Mar 04 2018
%F A289366 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 A289366 a(n) * A289365(n) ~ -sin(Pi/144) * exp(4*Pi*n) / (144*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A289366 nmax = 20; CoefficientList[Series[((1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3)^(1/288), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 08 2017 *)
%Y A289366 (E_6^2/E_4^3)^(k/288): this sequence (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), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).
%Y A289366 Cf. A000521 (j), A066395 (1/j), A289365, A289367, A300025, A305886.
%K A289366 sign
%O A289366 0,2
%A A289366 _Seiichi Manyama_, Jul 04 2017