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

0,2

Links

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

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

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 *)

Prog

(PARI) x='x+O('x^30); Vec(serlaplace(exp(exp(x)-1+x)/(1-x) )) \\ G. C. Greubel, May 23 2018
(Magma) m:=25; R<x>:=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

Crossrefs

Cf. A113059.

Sequence in context: A367874 A317170 A127087 * A186374 A187249 A105151

Adjacent sequences: A113057 A113058 A113059 * A113061 A113062 A113063

Keyword

nonn

Author

Karol A. Penson, Oct 13 2005

Status

approved