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a(n) = n! * Sum_{k=1..n} sigma_3(k)/k.
3

%I #15 Aug 07 2022 04:44:41

%S 1,11,89,794,6994,72204,753108,8973264,111281616,1524322080,

%T 21601104480,340803192960,5483287025280,96044874750720,

%U 1748238132614400,34093033838438400,682396164763084800,14706429413353574400,323342442475011993600,7585740483060676608000

%N a(n) = n! * Sum_{k=1..n} sigma_3(k)/k.

%F E.g.f.: (1/(1-x)) * Sum_{k>0} x^k * (1 + x^k)/(k * (1 - x^k)^3).

%F E.g.f.: -(1/(1-x)) * Sum_{k>0} k^2 * log(1 - x^k).

%F a(n) ~ n! * Pi^4 * n^3 / 270. - _Vaclav Kotesovec_, Aug 07 2022

%t Table[n! * Sum[DivisorSigma[3, k]/k, {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Aug 07 2022 *)

%o (PARI) a(n) = n!*sum(k=1, n, sigma(k, 3)/k);

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*(1+x^k)/(k*(1-x^k)^3))/(1-x)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k^2*log(1-x^k))/(1-x)))

%Y Cf. A001158, A064603, A356010, A356297, A356298.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 03 2022