A296652 Coefficients in expansion of (E_6^2/E_4^3)^(1/72).

1, -24, -2592, -1525536, -499930368, -233042911056, -99547207597440, -46277719207526208, -21444241881136232448, -10206632934331485363576, -4897739115250118143468992, -2379385980983995218900931680, -1164826509542958652906666171392

Offset: 0

Author

Seiichi Manyama, Feb 15 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), this sequence (k=4), A297021 (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).

Cf. A000521 (j).

Programs

Mathematica

terms = 13;
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)^(1/72) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/72).

a(n) ~ -Gamma(1/4)^(1/9) * exp(2*Pi*n) / (12 * 2^(1/9) * 3^(71/72) * Pi^(1/12) * Gamma(35/36) * n^(37/36)). - Vaclav Kotesovec, Mar 04 2018

a(n) * A299697(n) ~ -sin(Pi/36) * exp(4*Pi*n) / (36*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018