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

1, -108, -7128, -5975856, -1648702944, -817564231656, -330392410226208, -154125342449733600, -69899495093389741824, -33019122368612611954332, -15654348707682435222420432, -7540807164973158284078993424

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), 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).

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)^(1/16) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/16), where j is the j-function.

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

a(n) * A299951(n) ~ -sin(Pi/8) * exp(4*Pi*n) / (8*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018