A263358 Expansion of Product_{k>=1} 1/(1-x^(k+2))^k.

1, 0, 0, 1, 2, 3, 5, 7, 12, 18, 29, 43, 69, 101, 155, 231, 347, 509, 759, 1106, 1626, 2359, 3428, 4938, 7127, 10194, 14587, 20756, 29498, 41716, 58932, 82888, 116413, 162924, 227602, 316988, 440696, 610953, 845469, 1167118, 1608178, 2210888, 3034124, 4155111

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2015

Keywords

• nonn

Comments

In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.

d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).

d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).

d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

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 (30000 terms)
• Eric Weisstein's World of Mathematics, Barnes G-Function

Crossrefs

Cf. A000219, A052847, A263359, A263360, A263361, A263362, A263363, A263364.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

Mathematica

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[E^Sum[x^(3*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ exp(1/12 - Pi^4/(108*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(1/36) * sqrt(Pi) / (A * sqrt(3) * Zeta(3)^(17/36) * n^(1/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.