A023874 Expansion of Product_{k>=1} (1 - x^k)^(-k^5).

1, 1, 33, 276, 1828, 12729, 88903, 582846, 3690325, 22864592, 138658796, 822374485, 4781447342, 27314310586, 153519181630, 849786024496, 4637270263913, 24970548655999, 132788838463944, 697863705334941, 3626864249759775, 18650694625385462, 94948991121030892

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..2338 (first 601 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.

Formula

a(n) ~ 3^(127/882) * (5*Zeta(7))^(127/1764) * exp(7 * n^(6/7) * (5*Zeta(7))^(1/7) / (2^(3/7) * 3^(5/7)) + Zeta'(-5)) / (2^(187/882) * n^(1009/1764) * sqrt(7*Pi)), where Zeta(7) = A013665 = 1.008349277381922826..., Zeta'(-5) = ((137/60 - gamma - log(2*Pi))/42 + 45*Zeta'(6) / (2*Pi^6))/6 = -0.0005729859801986352... . - Vaclav Kotesovec, Feb 27 2015

G.f.: exp( Sum_{n>=1} sigma_6(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017

a(n) = (1/n)*Sum_{k=1..n} sigma_6(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^5, 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 = 22; Series[ Product[1/(1 - x^k)^k^5, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

Prog

(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^5)) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Crossrefs

Column k=5 of A144048.

Sequence in context: A008515 A179995 A000539 * A265839 A257450 A301549

Adjacent sequences: A023871 A023872 A023873 * A023875 A023876 A023877

Keyword

nonn

Author

Olivier Gérard

Extensions

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

Status

approved