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A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

%I A023875 #47 Jun 27 2025 23:07:18
%S A023875 1,1,65,794,6970,69251,689896,6309849,55654858,483526120,4104495070,
%T A023875 33968248260,275366110929,2192975727284,17169583920204,
%U A023875 132264358228507,1003715206329332,7511468689508580,55479733165442038,404709688656248024,2917717129031507178
%N A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).
%H A023875 Seiichi Manyama, <a href="/A023875/b023875.txt">Table of n, a(n) for n = 0..1771</a> (first 451 terms from Alois P. Heinz)
%H A023875 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 A023875 Vaclav Kotesovec, <a href="/A023875/a023875.jpg">Graph - The asymptotic ratio for 10000 terms</a>
%H A023875 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 A023875 a(n) ~ exp(Pi * 2^(27/8) * n^(7/8) / (7*15^(1/8)) - 45*Zeta(7) / (8*Pi^6)) / (2^(29/16) * 15^(1/16) * n^(9/16)), where Zeta(7) = A013665 = 1.00834927738192... . - _Vaclav Kotesovec_, Feb 27 2015
%F A023875 G.f.: exp( Sum_{n>=1} sigma_7(n)*x^n/n ). - _Seiichi Manyama_, Mar 05 2017
%F A023875 a(n) = (1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k). - _Seiichi Manyama_, Mar 05 2017
%p A023875 with(numtheory):
%p A023875 a:= proc(n) option remember; `if`(n=0, 1,
%p A023875       add(add(d*d^6, d=divisors(j)) *a(n-j), j=1..n)/n)
%p A023875     end:
%p A023875 seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 02 2012
%t A023875 max = 20; Series[ Product[1/(1 - x^k)^k^6, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* _Jean-François Alcover_, Mar 05 2013 *)
%o A023875 (PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^6)) \\ _G. C. Greubel_, Oct 31 2018
%o A023875 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^6: k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A023875 Column k=6 of A144048.
%K A023875 nonn
%O A023875 0,3
%A A023875 _Olivier Gérard_
%E A023875 Definition corrected by _Franklin T. Adams-Watters_ and _R. J. Mathar_, Dec 04 2006