A113060 a(n) = n!*Sum_{k=0..n} bell(k+1)/k!, n=0,1..., where bell(n) are the Bell numbers, cf. A000110.

1, 3, 11, 48, 244, 1423, 9415, 70045, 581507, 5349538, 54173950, 600127047, 7229169001, 94170096335, 1319764307235, 19806944750672, 316993980880556, 5389579751775611, 97018268274166055

Offset: 0

Author

Karol A. Penson, Oct 13 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A113059.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1+x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

Mathematica

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1 + x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)
Table[n!Sum[BellB[k+1]/k!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, May 03 2020 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(exp(x)-1+x)/(1-x) )) \\ G. C. Greubel, May 23 2018

Formula

a(n) = (-1)^n*n!*Sum_{p >=1} LaguerreL(n, -n-1, p)/(p-1)!/exp(1), n>=0.

E.g.f.: exp(exp(x)-1+x)/(1-x).

Representation as the n-th moment of a positive weight function on a positive half-axis: The weight function is a piecewise continuous function which is a weighted infinite sum of shifted exponential distributions, in Maple notation: a(n)=int(x^n*sum(exp(p-x)*Heaviside(x-p)/(p-1)!, p=1..infinity))/(exp(1)), n=0, 1...

a(n) ~ exp(exp(1)) * n!. - Vaclav Kotesovec, Jun 26 2022