A023878 Expansion of Product_{k>=1} (1 - x^k)^(-k^9).

1, 1, 513, 20196, 413668, 12444489, 372960863, 9158023846, 223763768245, 5567490203192, 132000248840652, 3018181447183141, 68165389692659690, 1512302997486058542, 32793035921825542778, 698432551205542941608, 14654522099892985823429, 302753023792981375706399

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Comments

In general, column m > 0 of A144048 is asymptotic to (Gamma(m+2)*Zeta(m+2))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * (Gamma(m+2)*Zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2)) + Zeta'(-m)) / (sqrt(2*Pi*(m+2)) * n^((m+3-2*Zeta(-m))/(2*m+4))). - Vaclav Kotesovec, Mar 01 2015

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..995 (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=9 of A144048. - Alois P. Heinz, Nov 02 2012

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^9: 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^9, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012

Mathematica

nmax=30; CoefficientList[Series[Product[1/(1-x^k)^(k^9),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 01 2015 *)

PARI

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^9)) \\ G. C. Greubel, Oct 31 2018

Formula

a(n) ~ 3^(67/363) * 5^(67/726) * (7*Zeta(11))^(67/1452) * exp(11 * 3^(4/11) * n^(10/11) * (7*Zeta(11))^(1/11) / (2^(3/11) * 5^(9/11)) + Zeta'(-9)) / (2^(95/726) * sqrt(11*Pi) * n^(793/1452)), where Zeta(11) = A013669 = 1.00049418860411946..., Zeta'(-9) = (5*(7129/2520 - gamma - log(2*Pi))/66 + 14175*Zeta'(10) / (2*Pi^10))/10 = 0.00313014531978857275492576829... . - Vaclav Kotesovec, Feb 27 2015

G.f.: exp( Sum_{n>=1} sigma_10(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017

a(n) = (1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Extensions

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