Voozh

A350725

a(n) = Sum_{k=0..n} k! * k^(n-k) * Stirling1(n,k).

1, 1, 1, -4, -2, 274, -3442, -12552, 2108664, -63083232, 87416112, 112192496976, -7487840132544, 174521224997040, 19793498724358032, -3109195219736188416, 209306170972547346816, 2973238556525799866496, -3013574861684426837113728, 456220653756733889826621696

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..19.

FORMULA

E.g.f.: Sum_{k>=0} log(1 + k*x)^k / k^k.

MATHEMATICA

a[0] = 1; a[n_] := Sum[k! * k^(n-k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2022 *)

PROG

(PARI) a(n) = sum(k=0, n, k!*k^(n-k)*stirling(n, k, 1));

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/k^k)))