A023877 Expansion of Product_{k>=1} (1 - x^k)^(-k^8).

1, 1, 257, 6818, 105250, 2175491, 44988020, 796565173, 13803604854, 240522266760, 4044067171130, 65769795259820, 1051279656603367, 16517653032316394, 254354069377336990, 3847172021760617755, 57300325471166205776, 840900188345961238222, 12164188625099191500782

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1170 (first 301 terms from Alois P. Heinz)
• G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
• 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, p. 21.

Crossrefs

Column k=8 of A144048.

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^8: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^8, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012

Mathematica

max = 18; Series[ Product[1/(1 - x^k)^k^8, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

PARI

m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^8)) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) ~ exp(5 * Pi * 2^(17/10) * n^(9/10) / (3^(21/10) * 11^(1/10)) + 315*Zeta(9)/(4*Pi^8)) / (2^(13/20) * sqrt(5) * 33^(1/20) * n^(11/20)), where Zeta(9) = A013667 = 1.0020083928260822144... . - Vaclav Kotesovec, Feb 27 2015

G.f.: exp( Sum_{n>=1} sigma_9(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017

a(n) = (1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Extensions

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006