%I
%S 1,2,6,23,107,587,3725,26952,219756,1998951,20105485,221838905,
%T 2666280457,34689290378,485840964614,7288997427755,116634438986227,
%U 1982868327635663,35692311974248093,678159760252918824,13563246929216611852,284828660383365005643
%N a(n) = n!*Sum_{k=0..n} Bell(k)/k! (cf. A000110).
%C Sequence was originally defined as an infinite sum involving generalized Laguerre polynomials: a(n)= ((1)^n*n!/exp(1))*Sum_{k>=0} LaguerreL(n,n1,k)/k!, n=0,1... . It appears in the problem of normal ordering of functions of boson operators.
%C a(n) is the number of ways to linearly order the elements in a (possibly empty) subset S of {1,2,...,n} and then partition the complement of S.  _Geoffrey Critzer_, Aug 07 2015
%H Robert Israel, <a href="/A101053/b101053.txt">Table of n, a(n) for n = 0..450</a>
%F E.g.f: exp(exp(x)1)/(1x).
%p with(combinat): a:=n>add(bell(j)*n!/j!,j=0..n): seq(a(n),n=0..20); # _Zerinvary Lajos_, Mar 19 2007
%t nn = 21; Range[0, nn]! CoefficientList[Series[Exp[(Exp[x]1)]/(1x), {x, 0, nn}], x] (* _Geoffrey Critzer_, Aug 07 2015 *)
%o (PARI) egf(s)=my(v=Vec(s),i); while(polcoeff(s,i)==0,i++); i; vector(i+#v,j,polcoeff(s,j+i)*(j+i)!)
%o egf(exp(exp(x)1)/(1x)) \\ _Charles R Greathouse IV_, Aug 07 2015
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(exp(x)1)/(1x) )) \\ _G. C. Greubel_, Mar 31 2019
%o (MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Exp(x)1)/(1x) )); [Factorial(n1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Mar 31 2019
%o (Sage) m = 30; T = taylor(exp(exp(x)1)/(1x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Mar 31 2019
%Y Cf. A000110.
%K nonn
%O 0,2
%A _Karol A. Penson_, Nov 29 2004
%E New definition from _Vladeta Jovovic_, Dec 01 2004
