A052871 Expansion of e.g.f. -LambertW(x/(-1+x)).

0, 1, 4, 27, 268, 3585, 60846, 1255471, 30535912, 855688833, 27148954330, 962037575631, 37659124454700, 1613921425656865, 75156944627712598, 3778932799275876495, 204039148080188427856, 11774630933193827543553

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 842
• Index entries for reversions of series

Crossrefs

Cf. A008297, A060356.

Programs

Maple

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

Mathematica

CoefficientList[Series[-LambertW[x/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Maxima

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

PARI

x='x+O('x^50); concat([0], Vec(serlaplace(- lambertw(x/(-1+x))) )) \\ G. C. Greubel, Nov 08 2017

Formula

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

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

a(n) ~ (1+exp(-1))^(n+1/2)*n^(n-1). - Vaclav Kotesovec, Sep 30 2013

From Seiichi Manyama, Sep 10 2024: (Start)

E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(A(x))).

E.g.f.: Series_Reversion( x / (x + exp(x)) ). (End)

Extensions

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013