A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 6, 4, 1, 2, 6, 10, 6, 2, 6, 14, 20, 8, 6, 14, 29, 30, 13, 14, 34, 54, 50, 22, 34, 66, 99, 74, 43, 72, 133, 166, 119, 82, 148, 242, 276, 182, 166, 286, 438, 442, 301, 316, 541, 744, 701, 494, 608, 976, 1255

Offset

0,10

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^(4*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(169/900) * exp(d51 - Pi^4/(10800*Zeta(3))+ Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (300 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(281/900) * 5^(169/450) * sqrt(3*Pi) * n^(619/900)), where d51 = A263178 = Integral_{x=0..infinity} exp(-4*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 1/(25*x^2) + 19/(300*x*exp(x)) = -0.1269958671388232529452705747311358056... .

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=4, 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^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A263142, A263143, A263144, A263178, A263145, A035528, A262876, A263136.

Sequence in context: A144528 A290694 A146164 * A051510 A361800 A340958

Adjacent sequences: A263138 A263139 A263140 * A263142 A263143 A263144

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved