A334986 a(n) = exp(n) * Sum_{k>=0} (-1)^k * n^(k-1) * k^(n-1) / k!.

1, -1, 2, -5, 9, 53, -1107, 12983, -116470, 560049, 8370713, -346902877, 7551856337, -117404648467, 913399734614, 22560135521007, -1393700803877939, 44331044030953865, -979905458659247779, 10462396536804802459, 367799071887303276422, -30046998012662824941947

Offset: 1

Author

Ilya Gutkovskiy, May 18 2020

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A030019, A052888, A292866.

Programs

Mathematica

Table[Sum[(-1)^k StirlingS2[n - 1, k] n^(k - 1), {k, 0, n - 1}], {n, 1, 22}]
Table[BellB[n - 1, -n]/n, {n, 1, 22}]

PARI

a(n)={sum(k=0, n-1, (-1)^k * stirling(n-1,k,2) * n^(k-1))} \\ Andrew Howroyd, May 18 2020

Formula

a(n) = Sum_{k=0..n-1} (-1)^k * Stirling2(n-1,k) * n^(k-1).

a(n) = BellPolynomial_(n-1)(-n) / n.