A052858 Expansion of e.g.f. log(-1/(-1+x*exp(x)-x)).

0, 0, 2, 3, 16, 65, 456, 3157, 28624, 276705, 3136240, 38531141, 528468744, 7837577761, 126588882616, 2194957583925, 40854219413536, 810192673705793, 17082845929433952, 381225135102420997

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 826

Programs

Maple

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

Mathematica

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

Maxima

a(n):=(n)!*sum((k-1)!*stirling2(n-k,k)/(n-k)!,k,1,n/2); /* Vladimir Kruchinin, Mar 22 2016 */

Formula

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

a(n) ~ (n-1)! * r^n, where r = 1.23997788765655... is the root of the equation log(1+r)=1/r. - Vaclav Kotesovec, Sep 29 2013

a(n) = n!*Sum_{k=1..n/2}((k-1)!*stirling2(n-k,k)/(n-k)!). - Vladimir Kruchinin, Mar 22 2016

a(0) = a(1) = 0; a(n) = n + Sum_{k=2..n-1} k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 21 2025

Extensions

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