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A289292 Coefficients in expansion of E_4^(1/2).

%I A289292 #30 Mar 05 2018 03:28:56
%S A289292 1,120,-6120,737760,-107249640,17385063120,-3014720249760,
%T A289292 547287510713280,-102701836021530600,19762301660609250840,
%U A289292 -3878226140959368843120,773209219953012480001440,-156173318001506652330786720,31888935085481430265623676560
%N A289292 Coefficients in expansion of E_4^(1/2).
%H A289292 Seiichi Manyama, <a href="/A289292/b289292.txt">Table of n, a(n) for n = 0..425</a>
%H A289292 R. S. Maier, <a href="http://arxiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29b).
%F A289292 G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/2).
%F A289292 a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 3*Gamma(1/3)^9 / (32*sqrt(2)*Pi^(13/2)) = 0.27646925986847687648926173728588572192308632719... - _Vaclav Kotesovec_, Jul 02 2017, updated Mar 05 2018
%F A289292 G.f.: 3F2(1/6, 1/2, 5/6; 1, 1; 1728/j) where j is the elliptic modular invariant (A000521). - _Seiichi Manyama_, Jul 07 2017
%t A289292 terms = 14;
%t A289292 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A289292 E4[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 23 2018 *)
%Y A289292 E_k^(1/2): A289291 (k=2), this sequence (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).
%Y A289292 E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), this sequence (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
%Y A289292 Cf. A001421, A004009 (E_4), A110163.
%K A289292 sign
%O A289292 0,2
%A A289292 _Seiichi Manyama_, Jul 02 2017