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A052848
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Number of ordered set partitions with a designated element in each block and no block containing less than two elements.
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21
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1, 0, 2, 3, 28, 125, 1146, 8827, 94200, 1007001, 12814390, 172114151, 2584755636, 41436880069, 721702509906, 13397081295795, 266105607506416, 5605474012933169, 125164378600050798, 2948082261121889983, 73122068527848758700, 1903894649651935410141
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(Maxima) a(n):=n!*sum((k!*stirling2(n-k, k))/(n-k)!, k, 0, n/2); /* Vladimir Kruchinin, Nov 16 2011 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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