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A299857 Coefficients in expansion of (E_6^2/E_4^3)^(1/16).

%I A299857 #18 Mar 04 2018 12:43:12
%S A299857 1,-108,-7128,-5975856,-1648702944,-817564231656,-330392410226208,
%T A299857 -154125342449733600,-69899495093389741824,-33019122368612611954332,
%U A299857 -15654348707682435222420432,-7540807164973158284078993424
%N A299857 Coefficients in expansion of (E_6^2/E_4^3)^(1/16).
%H A299857 Seiichi Manyama, <a href="/A299857/b299857.txt">Table of n, a(n) for n = 0..367</a>
%F A299857 G.f.: (1 - 1728/j)^(1/16), where j is the j-function.
%F A299857 a(n) ~ -3^(1/16) * sqrt(Gamma(1/4)) * exp(2*Pi*n) / (8 * sqrt(2) * Pi^(3/8) * Gamma(7/8) * n^(9/8)). - _Vaclav Kotesovec_, Mar 04 2018
%F A299857 a(n) * A299951(n) ~ -sin(Pi/8) * exp(4*Pi*n) / (8*Pi*n^2). - _Vaclav Kotesovec_, Mar 04 2018
%t A299857 terms = 12;
%t A299857 E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
%t A299857 E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
%t A299857 (E6[x]^2/E4[x]^3)^(1/16) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 28 2018 *)
%Y A299857 (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), A299856 (k=16), this sequence (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 A299857 Cf. A000521 (j).
%K A299857 sign
%O A299857 0,2
%A A299857 _Seiichi Manyama_, Feb 21 2018