A101053 - OEIS

%I #33 Jul 14 2022 05:17:42

%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,-n-1,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)/(1-x).

%F a(n) ~ exp(exp(1)-1) * n!. - _Vaclav Kotesovec_, Jun 26 2022

%F a(0) = 1; a(n) = Sum_{k=1..n} ((k-1)! + 1) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Jul 14 2022

%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)]/(1-x), {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)/(1-x)) \\ _Charles R Greathouse IV_, Aug 07 2015

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(exp(x)-1)/(1-x) )) \\ _G. C. Greubel_, Mar 31 2019

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ((j-1)!+1)*binomial(i-1, j-1)*v[i-j+1])); v; \\ _Seiichi Manyama_, Jul 14 2022

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Exp(x)-1)/(1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Mar 31 2019

%o (Sage) m = 30; T = taylor(exp(exp(x)-1)/(1-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Mar 31 2019

%Y Cf. A000110, A186755, A278677, A352270.

%K nonn

%O 0,2

%A _Karol A. Penson_, Nov 29 2004

%E New definition from _Vladeta Jovovic_, Dec 01 2004