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A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

%I A319647 #33 Oct 28 2018 03:48:29
%S A319647 1,1,6,38,526,13074,702813,70939556,13879861574,5583837482767,
%T A319647 4393101918607162,6717450870069292051,21057681806321501744772,
%U A319647 131246096280071506595491449,1604095619160115980216291007253,40299198842857238408636666363954678,2031474817845087309816967328335309651478
%N A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).
%H A319647 Seiichi Manyama, <a href="/A319647/b319647.txt">Table of n, a(n) for n = 0..80</a>
%F A319647 a(n) = [x^n] Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^n).
%F A319647 a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^k))).
%p A319647 with(numtheory):
%p A319647 b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p A319647       sigma[k](d), d=divisors(j))*b(n-j, k), j=1..n)/n)
%p A319647     end:
%p A319647 a:= n-> b(n$2):
%p A319647 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 26 2018
%t A319647 Table[SeriesCoefficient[Product[1/(1 - x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
%t A319647 Table[SeriesCoefficient[Product[Product[1/(1 - x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
%t A319647 Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]
%o A319647 (PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^sigma(k, n)), n)} \\ _Seiichi Manyama_, Oct 27 2018
%Y A319647 Cf. A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547, A321042.
%K A319647 nonn
%O A319647 0,3
%A A319647 _Ilya Gutkovskiy_, Oct 26 2018