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%I #13 Sep 08 2022 08:45:23
%S 1,3,11,48,244,1423,9415,70045,581507,5349538,54173950,600127047,
%T 7229169001,94170096335,1319764307235,19806944750672,316993980880556,
%U 5389579751775611,97018268274166055
%N a(n) = n!*Sum_{k=0..n} bell(k+1)/k!, n=0,1..., where bell(n) are the Bell numbers, cf. A000110.
%H G. C. Greubel, <a href="/A113060/b113060.txt">Table of n, a(n) for n = 0..448</a>
%F a(n) = (-1)^n*n!*Sum_{p >=1} LaguerreL(n, -n-1, p)/(p-1)!/exp(1), n>=0.
%F E.g.f.: exp(exp(x)-1+x)/(1-x).
%F 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...
%F a(n) ~ exp(exp(1)) * n!. - _Vaclav Kotesovec_, Jun 26 2022
%t 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 *)
%t Table[n!Sum[BellB[k+1]/k!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, May 03 2020 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(exp(x)-1+x)/(1-x) )) \\ _G. C. Greubel_, May 23 2018
%o (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
%Y Cf. A113059.
%K nonn
%O 0,2
%A _Karol A. Penson_, Oct 13 2005