A299994 Coefficients in expansion of (E_4^3/E_6^2)^(1/8).

1, 216, 49248, 21609504, 9000122112, 4129083886032, 1919370450227328, 917374442680570176, 444151666318727522304, 217813424092164713883960, 107771776495186976967396672, 53736084111333058216805911392, 26958647064591216695092188902400

Offset: 0

Author

Seiichi Manyama, Feb 22 2018

Keywords

• nonn

Links

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

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), this sequence (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299859.

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}];
(E4[x]^3/E6[x]^2)^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299859.

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

a(n) * A299859(n) ~ -exp(4*Pi*n) / (4*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018