A334258 a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.

1, 1, 1, -5, -74, -679, -4899, -17289, 325837, 10627109, 199348590, 2684041427, 15872610469, -546948563407, -27499774835519, -778467357484561, -15311413773551790, -125363405319188419, 6452292137017871097, 436442148982835915339, 16494863323310244977581

Offset: 1

Author

Ilya Gutkovskiy, Apr 20 2020

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A002741, A052888, A292866.

Programs

Mathematica

nmax = 21; CoefficientList[InverseSeries[Series[-Log[1 - x] Exp[-x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[Sum[(-1)^(n - k) StirlingS2[n, k] n^(k - 1), {k, 1, n}], {n, 1, 21}]
Table[(-1)^n BellB[n, -n]/n, {n, 1, 21}]

PARI

a(n) = sum(k=1, n, (-1)^(n-k) * stirling(n,k,2) * n^(k-1)); \\ Michel Marcus, Apr 20 2020

Formula

E.g.f.: series reversion of -log(1 - x) * exp(-x).

a(n) = (n - 1)! * [x^n] exp(n*(1 - exp(-x))).

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

a(n) = (-1)^n * BellPolynomial_n(-n) / n.