A052868 Expansion of e.g.f. LambertW(x/(-1+x))/x*(-1+x).

1, 1, 5, 40, 449, 6556, 118507, 2561518, 64540625, 1859206600, 60309007091, 2176222795594, 86488677518905, 3754431762036892, 176771908657345835, 8973513955735900246, 488586200931213192353, 28404347922603101834512

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..369
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 839.

Crossrefs

Cf. A352410, A370436.

Programs

Maple

spec := [S,{C=Sequence(Z,1 <= card),S=Set(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

CoefficientList[Series[LambertW[x/(-1+x)]/x*(-1+x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
nmax = 20; A[] = 0; Do[A[x] = Product[Exp[x^k*A[x]], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 01 2024 *)

Maxima

makelist(if n=0 then 1 else sum(n!/k!*binomial(n-1, k-1)*(k+1)^(k-1),k,0,n),n,0,17);  /* Bruno Berselli, May 25 2011 */

PARI

x='x+O('x^50); Vec(serlaplace(lambertw(x/(-1+x))/x*(-1+x))) \\ G. C. Greubel, Nov 12 2017

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x))))) \\ Seiichi Manyama, Mar 01 2023

Formula

E.g.f.: LambertW(x/(-1+x))/x*(-1+x).

a(n) = Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*(k+1)^(k-1). - Vladeta Jovovic, Sep 17 2003

a(n) ~ sqrt((exp(1)+1)*exp(1))*n^(n-1)*(1+exp(-1))^n. - Vaclav Kotesovec, Sep 29 2013

E.g.f. A(x) satisfies A(x) = exp( x*A(x)/(1-x) ) - Olivier Gérard, Dec 28 2013

E.g.f.: exp( -LambertW(-x/(1-x)) ). - Seiichi Manyama, Mar 01 2023

Extensions

New name using e.g.f., Joerg Arndt, Sep 30 2013