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A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3). 20

%I

%S 1,1,9,36,136,477,1703,5746,19099,61622,195366,607069,1856516,5586870,

%T 16579850,48549116,140438966,401592524,1136121837,3181700219,

%U 8825733603,24261363403,66124058839,178757752892,479513547399,1276792213203,3375707760306,8864712158225

%N Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

%H Seiichi Manyama, <a href="/A023872/b023872.txt">Table of n, a(n) for n = 0..5422</a> (first 1001 terms from Alois P. Heinz)

%H G. Almkvist, <a href="http://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

%F 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

%F G.f.: exp( Sum_{n>=1} sigma_4(n)*x^n/n ). - _Seiichi Manyama_, Mar 04 2017

%F a(n) = (1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1,

%p add(add(d*d^3, d=divisors(j)) *a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 02 2012

%t max = 27; Series[ Product[ 1/(1-x^k)^k^3, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Mar 05 2013 *)

%o (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

%o (MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^3: k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018

%o (SageMath) # uses[EulerTransform from A166861]

%o b = EulerTransform(lambda n: n^3)

%o print([b(n) for n in range(30)]) # _Peter Luschny_, Nov 11 2020

%Y Column k=3 of A144048.

%K nonn

%O 0,3

%A _Olivier Gérard_

%E Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006

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Last modified April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)