A289308 Coefficients in expansion of E_4^(3/8).

1, 90, -5940, 758520, -115431930, 19355028840, -3447208777320, 639751846440960, -122326632902618100, 23925871041887048130, -4763590542726586318440, 962102309316632909723880, -196619722885250960565506040, 40580696990507644723354537320

Offset: 0

Author

Seiichi Manyama, Jul 02 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..424

Crossrefs

E_4^(k/8): A108091 (k=1), A289307 (k=2), this sequence (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

Programs

Mathematica

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(3/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(3*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 3^(7/4) * Gamma(1/3)^(27/4) / (64 * 2^(3/8) * Pi^(9/2) * Gamma(5/8)) = 0.2574920621515873836544977885672468081360882154861344422709504189964... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018