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A023877 Expansion of Product_{k>=1} (1 - x^k)^(-k^8).

%I A023877 #45 Sep 08 2022 08:44:48
%S A023877 1,1,257,6818,105250,2175491,44988020,796565173,13803604854,
%T A023877 240522266760,4044067171130,65769795259820,1051279656603367,
%U A023877 16517653032316394,254354069377336990,3847172021760617755,57300325471166205776,840900188345961238222,12164188625099191500782
%N A023877 Expansion of Product_{k>=1} (1 - x^k)^(-k^8).
%H A023877 Seiichi Manyama, <a href="/A023877/b023877.txt">Table of n, a(n) for n = 0..1170</a> (first 301 terms from Alois P. Heinz)
%H A023877 G. Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H A023877 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 A023877 a(n) ~ exp(5 * Pi * 2^(17/10) * n^(9/10) / (3^(21/10) * 11^(1/10)) + 315*Zeta(9)/(4*Pi^8)) / (2^(13/20) * sqrt(5) * 33^(1/20) * n^(11/20)), where Zeta(9) = A013667 = 1.0020083928260822144... . - _Vaclav Kotesovec_, Feb 27 2015
%F A023877 G.f.: exp( Sum_{n>=1} sigma_9(n)*x^n/n ). - _Seiichi Manyama_, Mar 05 2017
%F A023877 a(n) = (1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k). - _Seiichi Manyama_, Mar 05 2017
%p A023877 with(numtheory):
%p A023877 a:= proc(n) option remember; `if`(n=0, 1,
%p A023877       add(add(d*d^8, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A023877     end:
%p A023877 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 02 2012
%t A023877 max = 18; Series[ Product[1/(1 - x^k)^k^8, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* _Jean-François Alcover_, Mar 05 2013 *)
%o A023877 (PARI) m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^8)) \\ _G. C. Greubel_, Oct 31 2018
%o A023877 (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^8: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A023877 Column k=8 of A144048.
%K A023877 nonn
%O A023877 0,3
%A A023877 _Olivier Gérard_
%E A023877 Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006