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A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

%I A301547 #13 Oct 26 2018 14:22:28
%S A301547 1,1,514,20198,414696,12465714,373679122,9181285000,224372879810,
%T A301547 5583837482767,132433701077938,3028947042351535,68425900639083569,
%U A301547 1518510622688185301,32936878700790531296,701684036762210944310,14726705417058058788172,304326729686784847885978
%N A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).
%H A301547 Seiichi Manyama, <a href="/A301547/b301547.txt">Table of n, a(n) for n = 0..995</a>
%F A301547 a(n) ~ exp((11*Pi)^(10/11) * (Zeta(11)/3)^(1/11) * n^(10/11) / (2^(3/11) * 5^(10/11)) - Zeta'(-9)/2) * (5*Zeta(11)/(3*Pi))^(131/2904) / (2^(131/968) * 11^(1583/2904) * n^(1583/2904)).
%F A301547 G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Oct 26 2018
%p A301547 with(numtheory):
%p A301547 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A301547       sigma[9](d), d=divisors(j))*a(n-j), j=1..n)/n)
%p A301547     end:
%p A301547 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 26 2018
%t A301547 nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A301547 Cf. A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8).
%Y A301547 Cf. A013957, A301553.
%K A301547 nonn
%O A301547 0,3
%A A301547 _Vaclav Kotesovec_, Mar 23 2018