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%I #18 Mar 05 2018 05:13:16
%S 1,210,-1260,232680,-28907970,4211355960,-671557897080,
%T 113817372354240,-20151698294479500,3687092782592216970,
%U -692109989731133096760,132609267059636375116920,-25838624519733523814390760,5105657091664960508653858680
%N Coefficients in expansion of E_4^(7/8).
%C In general, for 0 < m < 1, the expansion of (E_4)^m is asymptotic to (-1)^(n+1) * m * 3^(2*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*n) / (2^(9*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - _Vaclav Kotesovec_, Mar 05 2018
%H Seiichi Manyama, <a href="/A289319/b289319.txt">Table of n, a(n) for n = 0..425</a>
%F G.f.: Product_{n>=1} (1-q^n)^(7*A110163(n)/8).
%F a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 7 * 3^(7/4) * Gamma(1/3)^(63/4) / (1024 * 2^(7/8) * Pi^(21/2) * Gamma(1/8)) = 0.1121182787986009012644546699220584282491804117887058146553161217384... - _Vaclav Kotesovec_, Jul 08 2017, updated Mar 05 2018
%t nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(7/8), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 08 2017 *)
%Y E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), this sequence (k=7).
%Y Cf. A004009 (E_4), A110163.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 02 2017
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