A023873 Expansion of Product_{k>=1} (1 - x^k)^(-k^4).

1, 1, 17, 98, 490, 2411, 11940, 56093, 256274, 1140980, 4977222, 21273772, 89281011, 368408970, 1496993290, 5996312751, 23700208290, 92510062036, 356887002352, 1361671469470, 5141380256124, 19221678032134, 71190778935805, 261320839754142, 951091521384860

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 7. Cf. A380581. - Peter Bala, Jan 21 2025

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3339 (first 801 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=4 of A144048.

Cf. A283271, A380581.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^4: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^4, 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 = 27; Series[ Product[1/(1 - x^k)^k^4, {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^4)) \\ G. C. Greubel, Oct 30 2018

Formula

a(n) ~ exp(Pi * 2^(3/2) * 3^(2/3) * n^(5/6) / (5 * 7^(1/6)) + 3*Zeta(5) / (4*Pi^4)) / (2^(3/4) * 3^(2/3) * 7^(1/12) * n^(7/12)), where Zeta(5) = A013663 = 1.036927755143369926... . - Vaclav Kotesovec, Feb 27 2015

G.f.: exp( Sum_{n>=1} sigma_5(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017

a(n) = (1/n)*Sum_{k=1..n} sigma_5(k)*a(n-k). - Seiichi Manyama, Mar 04 2017

Extensions

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