A299863 Coefficients in expansion of (E_6^2/E_4^3)^(1/9).

1, -192, -4608, -9494784, -1988603904, -1136127187584, -419383041398784, -200225564597488128, -88040635024586342400, -41470393697874515307456, -19381646100387803980004352, -9267227811160245194038205184

Offset: 0

Author

Seiichi Manyama, Feb 21 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..367

Crossrefs

(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), A299857 (k=18), A299858 (k=24), this sequence (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), A289210 (k=288).

Cf. A000521 (j).

Programs

Mathematica

terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]^2/E4[x]^3)^(1/9) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/9), where j is the j-function.

a(n) ~ -2^(1/9) * Gamma(1/4)^(8/9) * exp(2*Pi*n) / (3^(17/9) * Pi^(2/3) * Gamma(7/9) * n^(11/9)). - Vaclav Kotesovec, Mar 04 2018

a(n) * A299993(n) ~ -2*sin(2*Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018