OFFSET
0,3
COMMENTS
a(n) is the number of functional digraphs on {1,2,...,n} such that no node is at a distance greater than one from a cycle (A006153) and every recurrent element has at least one nonrecurrent element mapped to it. - Geoffrey Critzer, Dec 07 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..434
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 816
FORMULA
E.g.f.: -1/(-1+x*exp(x)-x).
a(n) = n!*Sum_{k=0..floor(n/2)} k!*Stirling2(n-k,k)/(n-k)!. - Vladimir Kruchinin, Nov 16 2011
a(n) ~ n!/(1+r+r^2) * r^(n+2), where r = 1.23997788765655... is the root of the equation log(1+r)=1/r. - Vaclav Kotesovec, Oct 05 2013
a(0) = 1; a(n) = n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 04 2023
MAPLE
spec := [S, {B=Prod(Z, C), C=Set(Z, 1 <= card), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-j)*binomial(n, j)*j, j=2..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, May 11 2016
MATHEMATICA
nn=20; a=x Exp[x]; First[Range[0, nn]! CoefficientList[Series[1/(1-x (Exp[x]-1+y)), {x, 0, nn}], {y, x}]] Range[0, nn]! (* Geoffrey Critzer, Dec 07 2012 *)
PROG
(Maxima) a(n):=n!*sum((k!*stirling2(n-k, k))/(n-k)!, k, 0, n/2); /* Vladimir Kruchinin, Nov 16 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Better name from Geoffrey Critzer, Dec 10 2012
STATUS
approved