A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).

1, -1, 3, -16, 121, -1181, 14114, -199543, 3257139, -60279214, 1247164055, -28525394481, 714681439212, -19465007759913, 572609747089735, -18093710202583480, 611202186074834221, -21979340746682042249, 838330656532184312218, -33803668628843391999843

Offset: 0

Author

Seiichi Manyama, Feb 05 2022

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A282190, A305981.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      (-m)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2022

Mathematica

Table[Sum[(-1)^k * k^k * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 05 2022 *)

PARI

a(n) = sum(k=0, n, (-k)^k*stirling(n, k, 2));

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(exp(x)-1))))

Formula

E.g.f.: 1/(1 + LambertW(exp(x) - 1)), where LambertW() is the Lambert W-function.

a(n) ~ (-1)^n * n^n / (sqrt(exp(1)-1) * (1 - log(exp(1)-1))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Feb 05 2022