A101053 a(n) = n! * Sum_{k=0..n} Bell(k)/k! (cf. A000110).

1, 2, 6, 23, 107, 587, 3725, 26952, 219756, 1998951, 20105485, 221838905, 2666280457, 34689290378, 485840964614, 7288997427755, 116634438986227, 1982868327635663, 35692311974248093, 678159760252918824, 13563246929216611852, 284828660383365005643

Offset

0,2

Comments

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.

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

Links

Robert Israel, Table of n, a(n) for n = 0..450

Formula

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

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

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

Maple

with(combinat): a:=n->add(bell(j)*n!/j!, j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Mar 19 2007

Mathematica

nn = 21; Range[0, nn]! CoefficientList[Series[Exp[(Exp[x]-1)]/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)

Prog

(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)!)
egf(exp(exp(x)-1)/(1-x)) \\ Charles R Greathouse IV, Aug 07 2015
(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(exp(x)-1)/(1-x) )) \\ G. C. Greubel, Mar 31 2019
(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
(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
(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

Crossrefs

Cf. A000110, A186755, A278677, A352270.

Sequence in context: A113226 A071075 A007555 * A155857 A071076 A297196

Adjacent sequences: A101050 A101051 A101052 * A101054 A101055 A101056

Keyword

nonn

Author

Karol A. Penson, Nov 29 2004

Extensions

New definition from Vladeta Jovovic, Dec 01 2004

Status

approved