A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

1, 1, 65, 794, 6970, 69251, 689896, 6309849, 55654858, 483526120, 4104495070, 33968248260, 275366110929, 2192975727284, 17169583920204, 132264358228507, 1003715206329332, 7511468689508580, 55479733165442038, 404709688656248024, 2917717129031507178

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1771 (first 451 terms from Alois P. Heinz)
• G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
• Vaclav Kotesovec, Graph - The asymptotic ratio for 10000 terms
• 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=6 of A144048.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^6: 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^6, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2012

Mathematica

max = 20; Series[ Product[1/(1 - x^k)^k^6, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

PARI

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^6)) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) ~ exp(Pi * 2^(27/8) * n^(7/8) / (7*15^(1/8)) - 45*Zeta(7) / (8*Pi^6)) / (2^(29/16) * 15^(1/16) * n^(9/16)), where Zeta(7) = A013665 = 1.00834927738192... . - Vaclav Kotesovec, Feb 27 2015

G.f.: exp( Sum_{n>=1} sigma_7(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017

a(n) = (1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Extensions

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