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A052848
Number of ordered set partitions with a designated element in each block and no block containing less than two elements.
28
1, 0, 2, 3, 28, 125, 1146, 8827, 94200, 1007001, 12814390, 172114151, 2584755636, 41436880069, 721702509906, 13397081295795, 266105607506416, 5605474012933169, 125164378600050798, 2948082261121889983, 73122068527848758700, 1903894649651935410141
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
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
Cf. A000296.
Sequence in context: A012697 A191470 A001094 * A357267 A074233 A354611
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Better name from Geoffrey Critzer, Dec 10 2012
STATUS
approved