A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

1, 0, 2, -13, 127, -1573, 23711, -421356, 8626668, -199971255, 5177291275, -148078588667, 4636966634653, -157786054331852, 5797411243015250, -228749440644895405, 9646951350227609155, -433035586385769361001, 20614401475233006857035, -1037331650810058231498688

Offset: 0

Author

Ilya Gutkovskiy, Oct 06 2017

Keywords

• sign

Comments

The n-th term of the n-th inverse binomial transform of A000110.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..386
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.

Main diagonal of A361781.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
[A290219(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Mathematica

Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]

SageMath

[factorial(n)*( exp(exp(x) -n*x -1) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

Formula

a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - Vaclav Kotesovec, Aug 04 2021