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%I A023878 #54 Sep 23 2022 16:18:28
%S A023878 1,1,513,20196,413668,12444489,372960863,9158023846,223763768245,
%T A023878 5567490203192,132000248840652,3018181447183141,68165389692659690,
%U A023878 1512302997486058542,32793035921825542778,698432551205542941608,14654522099892985823429,302753023792981375706399
%N A023878 Expansion of Product_{k>=1} (1 - x^k)^(-k^9).
%C A023878 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
%H A023878 Seiichi Manyama, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwMjM4NzgvYjAyMzg3OC50eHQ">Table of n, a(n) for n = 0..995</a> (first 301 terms from Alois P. Heinz)
%H A023878 G. Almkvist, <a href="/load/view.php?a=aHR0cHM6Ly9wcm9qZWN0ZXVjbGlkLm9yZy9ldWNsaWQuZW0vMTA0NzY3NDE1Mg">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H A023878 Vaclav Kotesovec, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMTUwOS4wODcwOA">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 A023878 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
%F A023878 G.f.: exp( Sum_{n>=1} sigma_10(n)*x^n/n ). - _Seiichi Manyama_, Mar 05 2017
%F A023878 a(n) = (1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k). - _Seiichi Manyama_, Mar 05 2017
%p A023878 with(numtheory):
%p A023878 a:= proc(n) option remember; `if`(n=0, 1,
%p A023878       add(add(d*d^9, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A023878     end:
%p A023878 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 02 2012
%t A023878 nmax=30; CoefficientList[Series[Product[1/(1-x^k)^(k^9),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 01 2015 *)
%o A023878 (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
%o A023878 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^9: k in [1..m]]) )); // _G. C. Greubel_, Oct 3012018
%Y A023878 Column k=9 of A144048. - _Alois P. Heinz_, Nov 02 2012
%K A023878 nonn
%O A023878 0,3
%A A023878 _Olivier Gérard_
%E A023878 Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006

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