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a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).
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%I #15 Aug 17 2022 03:08:12

%S 1,7,38,240,1509,12115,96326,929432,9421089,108909943,1249105054,

%T 17862483320,241674418101,3676733397363,59149265744302,

%U 1058605924855568,18041587282787489,363409114370324295,6970858463185187062,153017341796727034336,3360005220780469981157

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

%C The average value of a(n) is zeta(3) * exp(1) * n * n!. - _Vaclav Kotesovec_, Aug 17 2022

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

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

%t Table[n! * Sum[DivisorSigma[2, k]/(k * (n-k)!), {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Aug 17 2022 *)

%o (PARI) a(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));

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

%Y Cf. A002745, A002746, A356589.

%Y Cf. A356298.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 15 2022