A368725 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / k!.

1, 1, 2, 9, 100, 1065, 10626, 224161, 4598504, 46288017, 2509940710, 84061763841, -1602021820596, 164372205860473, 5216105126641514, -883395389739028095, 79008645559978113616, -1023235751229436800735, -651030746777115881959602, 113943411938145511923004513

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial.
• Wikipedia, Touchard polynomials.

Crossrefs

Main diagonal of A368724.

Cf. A000587, A256016.

Programs

Mathematica

Join[{1}, Table[n!*Sum[(-1)^(n-k)*k^n/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 18 2025 *)

PARI

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

Formula

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

a(n) = n! * [x^n] B_n(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

a(n) ~ A000587(n) * (-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025