A262946 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^(3*k-1).

1, 0, 2, 0, 3, 5, 4, 10, 13, 15, 37, 31, 61, 87, 99, 178, 228, 286, 477, 552, 816, 1163, 1418, 2077, 2790, 3507, 5113, 6478, 8563, 11888, 15005, 20100, 27054, 34055, 46002, 59905, 76436, 102105, 130879, 168103, 221954, 281300, 363743, 472557, 597579, 772148

Offset: 0

Author

Vaclav Kotesovec, Oct 05 2015

Keywords

• nonn

Comments

A262946(n)/A262947(n) ~ exp(3*(d1-d2)) * Gamma(1/3)^3 / (2*Pi)^(3/2), where d1 = A263030 and d2 = A263031. - Vaclav Kotesovec, Oct 08 2015

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• 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
• Vaclav Kotesovec, Graph - The asymptotic ratio (120000 terms)

Crossrefs

Cf. A035386, A262811, A262876, A262883, A262923, A262947.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, 3*r-1, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

Mathematica

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

Formula

a(n) ~ (2*Zeta(3))^(5/36) * exp(3*d1 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(29/36) * Gamma(2/3) * n^(23/36)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.18870819197952853237641009864920797359211446726842922150941... . - Vaclav Kotesovec, Oct 08 2015