A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 2, 3, 1, 2, 6, 5, 2, 6, 11, 7, 6, 15, 21, 12, 15, 30, 34, 22, 35, 58, 59, 43, 70, 108, 95, 85, 142, 187, 161, 167, 263, 318, 274, 318, 480, 534, 471, 595, 836, 879, 819, 1081, 1433, 1442, 1429, 1915, 2391, 2365, 2483, 3314, 3947

Offset

0,8

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Formula

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

a(n) ~ Zeta(3)^(53/288) * exp(d41 - Pi^4/(6912*Zeta(3)) + Pi^2 * n^(1/3) / (48*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (sqrt(3*Pi) * 2^(101/96) * n^(197/288)), where d41 = A263176 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 1/(16*x^2) + 5/(96*x*exp(x)) = -0.158924147180165035059952001737321408554746599955833696821824808027... .

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A035528, A262876, A263141, A263137, A263176, A263138.

Sequence in context: A071433 A110657 A071512 * A080018 A079686 A005813

Adjacent sequences: A263133 A263134 A263135 * A263137 A263138 A263139

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved