A289247 Coefficients in expansion of 1/E_4^(1/8).

1, -30, 3780, -616440, 111056910, -21135698280, 4165203862440, -840914061328320, 172810940671692900, -35998781800053352710, 7579904611028433074280, -1609957152292592382408360, 344417407415742189796786680, -74127324674775434904036905640

Offset: 0

Author

Seiichi Manyama, Jul 08 2017

Keywords

• sign

Links

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

Crossrefs

E_4^(k/8): A001943 (k=-8), A289566 (k=-4), A295815 (k=-2), this sequence (k=-1), A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8).

Cf. A110163, A300147.

Programs

Mathematica

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

Formula

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

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = Pi^(3/2) / (2^(15/8) * 3^(1/4) * Gamma(1/3)^(9/4) * Gamma(9/8)) = 0.133402757019143151407904538533... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018