A317362 Expansion of e.g.f. exp(exp(x/(1 + x)) - 1).

1, 1, 0, -1, 3, -8, 23, -89, 556, -4773, 44425, -397670, 3060577, -12448655, -235761640, 9571505555, -241952653453, 5424619822460, -116900288145113, 2494797839905055, -53406941947725348, 1152770311462756071, -25109138533156554399, 550613923917090815374, -12088287036694435407999

Offset: 0

Author

Ilya Gutkovskiy, Jul 26 2018

Keywords

• sign

Comments

Inverse Lah transform of the Bell numbers (A000110).

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000110, A084357.

Programs

Maple

a:= proc(n) option remember; add((-1)^(n-k)*n!/k!*
      binomial(n-1, k-1)*combinat[bell](k), k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

Mathematica

nmax = 24; CoefficientList[Series[Exp[Exp[x/(1 + x)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] BellB[k] n!/k!, {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*Bell(k)*n!/k!, where Bell() = A000110.