A263405 - OEIS

A263405

Expansion of Product_{k>=1} 1/(1-x^(3*k+1))^k.

1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 2, 1, 4, 6, 2, 6, 10, 6, 8, 20, 14, 13, 30, 29, 22, 50, 54, 43, 74, 99, 76, 119, 166, 144, 182, 276, 254, 294, 442, 451, 468, 701, 758, 772, 1088, 1270, 1256, 1698, 2052, 2067, 2618, 3294, 3352, 4065, 5162, 5430, 6284, 8050

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OFFSET

0,8

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

FORMULA

G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^(3*k))^2)).

a(n) ~ c * Zeta(3)^(19/108) * exp(-Pi^4/(3888*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), where c = 3^(1/6) * sqrt(2*Pi) * exp(A263031) / Gamma(1/3) = 1.107474840397395849254161220076423560365022...

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

`if`(irem(d-1, 3)=0, (d-1)/3, 0),

d=divisors(j))*a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..60); # after Alois P. Heinz, Oct 17 2015

MATHEMATICA

nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 60; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A262876, A263031, A263406, A263414, A263415.