A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

1, 0, 0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 6, 7, 10, 9, 19, 14, 29, 23, 46, 38, 66, 64, 99, 107, 143, 171, 211, 270, 311, 418, 465, 633, 698, 945, 1049, 1399, 1579, 2052, 2364, 2997, 3527, 4366, 5219, 6339, 7686, 9197, 11234, 13321, 16340, 19261, 23622, 27796

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2015

Keywords

• nonn

Comments

From Vaclav Kotesovec, Oct 17 2015: (Start)

In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.

d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).

d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).

d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

(End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• 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
• Eric Weisstein's World of Mathematics, Barnes G-Function.

Crossrefs

Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d>1 and d::odd, (d-3)/2, 0),
      d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

Mathematica

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

Formula

a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.