A334241 a(n) = exp(n) * Sum_{k>=0} (k + 1)^n * (-n)^k / k!.

1, 0, -1, 7, -43, 221, -341, -15980, 370761, -5688125, 62689871, -197586839, -14973562979, 585250669316, -14306382821485, 240985102271971, -1121421968408303, -122020498882279931, 6674724196051810807, -223424819176020519168, 5051515662105879438501

Offset: 0

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A292866, A334193, A334240, A334242, A334243.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[x + n (1 - Exp[x])], {x, 0, n}], {n, 0, 20}]
Table[Sum[Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (-n*x/(1 - x))^k / Product_{j=1..k} (1 - j*x/(1 - x)).

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

a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n).