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A297021 Coefficients in expansion of (E_6^2/E_4^3)^(1/48).

%I A297021 #38 Mar 04 2018 12:41:19
%S A297021 1,-36,-3672,-2240784,-719628768,-337401534456,-143188210269216,
%T A297021 -66549102831096480,-30753876262814297856,-14619380361359418716724,
%U A297021 -7003704012123711964880592,-3398241529278572532519050928,-1661531038403129009358413705856
%N A297021 Coefficients in expansion of (E_6^2/E_4^3)^(1/48).
%H A297021 Seiichi Manyama, <a href="/A297021/b297021.txt">Table of n, a(n) for n = 0..367</a>
%F A297021 G.f.: (1 - 1728/j)^(1/48).
%F A297021 a(n) ~ -Gamma(1/4)^(1/6) * exp(2*Pi*n) / (8 * 2^(1/6) * 3^(47/48) * Pi^(1/8) * Gamma(23/24) * n^(25/24)). - _Vaclav Kotesovec_, Mar 04 2018
%F A297021 a(n) * A299698(n) ~ -sin(Pi/24) * exp(4*Pi*n) / (24*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A297021 terms = 13;
%t A297021 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A297021 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A297021 (E6[x]^2/E4[x]^3)^(1/48) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)
%Y A297021 (E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), this sequence (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 A297021 Cf. A000521 (j).
%K A297021 sign
%O A297021 0,2
%A A297021 _Seiichi Manyama_, Feb 15 2018