A335529 a(n) = n! * [x^n] (1 - (n-1)*log(1 + x))/(1 - n*log(1 + x)).

1, 1, 3, 38, 1042, 49774, 3661128, 383653080, 54275300112, 9964363066848, 2303245150868640, 654457584668128416, 224205104879416320768, 91129285853151907958544, 43356207229026959513863680, 23868203329368882698589532800, 15053662436260897659550535387136

Offset: 0

Author

Seiichi Manyama, Jun 12 2020

Keywords

• nonn

Crossrefs

Main diagonal of A334369.

Cf. A317172, A321189.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[k! * n^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 12 2020 *)

PARI

{a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 1)))}

Formula

a(n) = A317172(n)/n = Sum_{k=0..n} k!*n^(k-1)*Stirling1(n,k) for n > 1.

a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 12 2020