A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3).
1, 1, 9, 36, 136, 477, 1703, 5746, 19099, 61622, 195366, 607069, 1856516, 5586870, 16579850, 48549116, 140438966, 401592524, 1136121837, 3181700219, 8825733603, 24261363403, 66124058839, 178757752892, 479513547399, 1276792213203, 3375707760306, 8864712158225
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..5422 (first 1001 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.
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^3: 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^3, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..30); # Alois P. Heinz, Nov 02 2012 -
Mathematica
max = 27; Series[ Product[ 1/(1-x^k)^k^3, {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^3)) \\ G. C. Greubel, Oct 30 2018 -
SageMath
# uses[EulerTransform from A166861] b = EulerTransform(lambda n: n^3) print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020
Formula
a(n) ~ (3*Zeta(5))^(59/600) * exp(5 * n^(4/5) * (3*Zeta(5))^(1/5) / 2^(7/5) + Zeta'(-3)) / (2^(41/200) * n^(359/600) * sqrt(5*Pi)), where Zeta(5) = A013663 = 1.036927755143369926..., Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.00537857635777430114441697421... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_4(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Extensions
Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006