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%I #41 Sep 08 2022 08:44:47
%S 1,1,33,276,1828,12729,88903,582846,3690325,22864592,138658796,
%T 822374485,4781447342,27314310586,153519181630,849786024496,
%U 4637270263913,24970548655999,132788838463944,697863705334941,3626864249759775,18650694625385462,94948991121030892
%N Expansion of Product_{k>=1} (1 - x^k)^(-k^5).
%H Seiichi Manyama, <a href="/A023874/b023874.txt">Table of n, a(n) for n = 0..2338</a> (first 601 terms from Alois P. Heinz)
%H G. Almkvist, <a href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">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^(127/882) * (5*Zeta(7))^(127/1764) * exp(7 * n^(6/7) * (5*Zeta(7))^(1/7) / (2^(3/7) * 3^(5/7)) + Zeta'(-5)) / (2^(187/882) * n^(1009/1764) * sqrt(7*Pi)), where Zeta(7) = A013665 = 1.008349277381922826..., Zeta'(-5) = ((137/60 - gamma - log(2*Pi))/42 + 45*Zeta'(6) / (2*Pi^6))/6 = -0.0005729859801986352... . - _Vaclav Kotesovec_, Feb 27 2015
%F G.f.: exp( Sum_{n>=1} sigma_6(n)*x^n/n ). - _Seiichi Manyama_, Mar 05 2017
%F a(n) = (1/n)*Sum_{k=1..n} sigma_6(k)*a(n-k). - _Seiichi Manyama_, Mar 05 2017
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(add(d*d^5, d=divisors(j)) *a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 02 2012
%t max = 22; Series[ Product[1/(1 - x^k)^k^5, {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^5)) \\ _G. C. Greubel_, Oct 30 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^5: k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018
%Y Column k=5 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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