A023876 G.f.: Product_{k>=1} (1 - x^k)^(-k^7).

1, 1, 129, 2316, 26956, 385017, 5512443, 70223666, 866470849, 10628564312, 126832407040, 1469751196093, 16694372607012, 186350644088784, 2042610304126944, 22007441766651756, 233482509248479425, 2441727926157182541, 25187101530316996950, 256456174925807404269

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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=7 of A144048.

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^7: 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^7, 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 = 19; Series[ Product[1/(1 - x^k)^k^7, {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^7)) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) ~ (35*Zeta(9))^(119/2160) * exp((3/2)^(20/9) * n^(8/9) * (35*Zeta(9))^(1/9) + Zeta'(-7)) / (2^(247/2160) * 3^(961/1080) * sqrt(Pi) * n^(1199/2160)), where Zeta(9) = A013667 = 1.0020083928260822144..., Zeta'(-7) = ((gamma + log(2*Pi) - 363/140)/30 - 315*Zeta'(8)/Pi^8)/8 = -0.00072864268015924... . - Vaclav Kotesovec, Feb 27 2015

Extensions

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