A052863 Expansion of e.g.f. log(-1/(-1+x))*exp(x) - log(-1/(-1+x)).

0, 0, 2, 6, 18, 65, 295, 1652, 11032, 85353, 749203, 7347384, 79564496, 942541041, 12121319327, 168145213732, 2502276609008, 39761200642225, 671855234838915, 12028625060491336, 227451564319791336, 4529507975800063337, 94751047516476943359, 2077192015403191663844

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 831

Programs

Maple

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

Mathematica

CoefficientList[Series[Log[-1/(-1+x)]*E^x-Log[-1/(-1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)

PARI

my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-log(1-x)*(exp(x)-1)))) \\ Seiichi Manyama, May 13 2022

PARI

a(n) = sum(k=0, n-2, k!*binomial(n, k+1)); \\ Seiichi Manyama, May 13 2022

Formula

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

Recurrence: {a(1)=0, a(3)=6, a(2)=2, (-n^3-2*n-3*n^2)*a(n)+(19*n+11*n^2+2*n^3+10)*a(n+1)+(-38*n-12*n^2-n^3-36)*a(n+2)+(41+26*n+4*n^2)*a(n+3)+(-17-5*n)*a(n+4)+2*a(n+5), a(4)=18, a(5)=65}

a(n) = A002104(n)-(n-1)!. - Vladeta Jovovic, Apr 03 2005

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

a(n) = Sum_{k=0..n-2} k! * binomial(n,k+1). - Seiichi Manyama, May 13 2022

Extensions

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