A289210 Coefficients in expansion of E_6^2/E_4^3.

1, -1728, 1285632, -616294656, 242544070656, -85253786824320, 27846073156184064, -8638345400999827968, 2579332695698905989120, -747814048389765750131136, 211795259563761765262894080, -58852853362216364363212075776

Offset: 0

Author

Seiichi Manyama, Jun 28 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..420

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), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), this sequence (k=288).

Cf. A000521 (j), A004009 (E_4), A013973 (E_6), A066395, A289209, A300025.

E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

Programs

Mathematica

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

a(n) = -1728 * A066395(n) for n > 0.

G.f.: 1 - 1728 * q * Product_{k>=1} (1-q^k)^24 / E_4^3 = 1 - 1728/j.

G.f.: (E_6*E_6)/(E_4*E_8) = (E_6*E_10)/(E_8*E_8). - Seiichi Manyama, Jun 29 2017

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n^2, where c = 256 * Pi^12 / Gamma(1/3)^18 = 4.684993039417145659090436569582265840407909701042523126716193567422... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018

a(0) = 1, a(n) = -(288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018