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%I #17 Aug 07 2022 04:45:18
%S 1,7,41,290,2074,18444,165108,1749264,19412496,241299360,3097006560,
%T 45546606720,673536159360,10986261431040,187460277177600,
%U 3445281394329600,64637392771123200,1325310849663897600,27498565425087590400,616389533324974080000
%N a(n) = n! * Sum_{k=1..n} sigma_2(k)/k.
%F E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
%F E.g.f.: -(1/(1-x)) * Sum_{k>0} k * log(1 - x^k).
%F a(n) ~ n! * zeta(3) * n^2 / 2. - _Vaclav Kotesovec_, Aug 07 2022
%t Table[n! * Sum[DivisorSigma[2, k]/k, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Aug 07 2022 *)
%o (PARI) a(n) = n!*sum(k=1, n, sigma(k, 2)/k);
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k*(1-x^k)^2))/(1-x)))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k*log(1-x^k))/(1-x)))
%Y Cf. A001157, A064602, A356010, A356297, A356323.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Aug 03 2022