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A299856 Coefficients in expansion of (E_6^2/E_4^3)^(1/18).

%I A299856 #17 Mar 04 2018 08:47:34
%S A299856 1,-96,-6912,-5410944,-1537640448,-753077521728,-307254291047424,
%T A299856 -143134552425743616,-65142005576276164608,-30798673631132393592288,
%U A299856 -14628259811568672073824768,-7054762801208507859522653568
%N A299856 Coefficients in expansion of (E_6^2/E_4^3)^(1/18).
%H A299856 Seiichi Manyama, <a href="/A299856/b299856.txt">Table of n, a(n) for n = 0..367</a>
%F A299856 G.f.: (1 - 1728/j)^(1/18), where j is the j-function.
%F A299856 a(n) ~ c * exp(2*Pi*n) / n^(10/9), where c = -Gamma(1/4)^(4/9) / (2^(4/9) * 3^(35/18) * Pi^(1/3) * Gamma(8/9)) = -0.0974650059642735838539936939997471425... - _Vaclav Kotesovec_, Mar 04 2018
%F A299856 a(n) * A299950(n) ~ -sin(Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A299856 terms = 12;
%t A299856 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A299856 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A299856 (E6[x]^2/E4[x]^3)^(1/18) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)
%Y A299856 (E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), this sequence (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 A299856 Cf. A000521 (j).
%K A299856 sign
%O A299856 0,2
%A A299856 _Seiichi Manyama_, Feb 21 2018