A299858 Coefficients in expansion of (E_6^2/E_4^3)^(1/12).

1, -144, -6912, -7563456, -1885022208, -979976901600, -383134788854784, -179914112738674560, -80649007527361757184, -38019764211792500474064, -17921855069499640580651520, -8604055343353988623666807872

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), this sequence (k=24), A299863 (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)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/12), where j is the j-function.

a(n) ~ -Gamma(1/4)^(2/3) * exp(2*Pi*n) / (2^(5/3) * 3^(11/12) * sqrt(Pi) * Gamma(5/6) * n^(7/6)). - Vaclav Kotesovec, Mar 04 2018

a(n) * A299953(n) ~ -exp(4*Pi*n) / (12*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018