cp's OEIS Frontend

A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

%I A318784 #8 Sep 03 2018 17:56:13
%S A318784 1,0,1,1,4,2,11,6,25,20,56,44,139,107,283,266,619,567,1317,1242,2680,
%T A318784 2705,5403,5539,10947,11339,21291,23013,41494,45213,79991,88312,
%U A318784 151546,170908,284901,324421,532505,611227,981002,1142000,1797451,2105773,3268765,3855050,5889704,7004451
%N A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).
%C A318784 Convolution of A061256 and A073592.
%C A318784 Euler transform of A001065.
%H A318784 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A318784 G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).
%F A318784 G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).
%F A318784 a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Sep 03 2018
%p A318784 with(numtheory):
%p A318784 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A318784       (sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)
%p A318784     end:
%p A318784 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 03 2018
%t A318784 nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A318784 nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A318784 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]
%Y A318784 Cf. A000203, A001065, A001157, A061256, A073592, A318783.
%K A318784 nonn
%O A318784 0,5
%A A318784 _Ilya Gutkovskiy_, Sep 03 2018