cp's OEIS Frontend

A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

%I A356600 #15 Aug 17 2022 03:08:12
%S A356600 1,7,38,240,1509,12115,96326,929432,9421089,108909943,1249105054,
%T A356600 17862483320,241674418101,3676733397363,59149265744302,
%U A356600 1058605924855568,18041587282787489,363409114370324295,6970858463185187062,153017341796727034336,3360005220780469981157
%N A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).
%C A356600 The average value of a(n) is zeta(3) * exp(1) * n * n!. - _Vaclav Kotesovec_, Aug 17 2022
%F A356600 E.g.f.: exp(x) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
%F A356600 E.g.f.: -exp(x) * Sum_{k>0} k * log(1 - x^k).
%t A356600 Table[n! * Sum[DivisorSigma[2, k]/(k * (n-k)!), {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Aug 17 2022 *)
%o A356600 (PARI) a(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));
%o A356600 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, x^k/(k*(1-x^k)^2))))
%o A356600 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, k*log(1-x^k))))
%Y A356600 Cf. A002745, A002746, A356589.
%Y A356600 Cf. A356298.
%K A356600 nonn
%O A356600 1,2
%A A356600 _Seiichi Manyama_, Aug 15 2022