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

0,3

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A008297, A060356.

Sequence in context: A353233 A265270 A161633 * A104653 A194787 A020558

Adjacent sequences: A052868 A052869 A052870 * A052872 A052873 A052874

Keyword

easy,nonn

Author

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

Extensions

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

Status

approved