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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.

%I A052848 #41 Dec 04 2023 06:35:42
%S A052848 1,0,2,3,28,125,1146,8827,94200,1007001,12814390,172114151,2584755636,
%T A052848 41436880069,721702509906,13397081295795,266105607506416,
%U A052848 5605474012933169,125164378600050798,2948082261121889983,73122068527848758700,1903894649651935410141
%N A052848 Number of ordered set partitions with a designated element in each block and no block containing less than two elements.
%C A052848 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
%H A052848 Alois P. Heinz, <a href="/A052848/b052848.txt">Table of n, a(n) for n = 0..434</a>
%H A052848 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=816">Encyclopedia of Combinatorial Structures 816</a>
%F A052848 E.g.f.: -1/(-1+x*exp(x)-x).
%F A052848 a(n) = n!*Sum_{k=0..floor(n/2)} k!*Stirling2(n-k,k)/(n-k)!. - _Vladimir Kruchinin_, Nov 16 2011
%F A052848 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
%F A052848 a(0) = 1; a(n) = n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Dec 04 2023
%p A052848 spec := [S,{B=Prod(Z,C),C=Set(Z,1 <= card),S=Sequence(B)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);
%p A052848 # second Maple program:
%p A052848 a:= proc(n) option remember; `if`(n=0, 1,
%p A052848       add(a(n-j)*binomial(n, j)*j, j=2..n))
%p A052848     end:
%p A052848 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 11 2016
%t A052848 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 *)
%o A052848 (Maxima) a(n):=n!*sum((k!*stirling2(n-k,k))/(n-k)!,k,0,n/2); /* _Vladimir Kruchinin_, Nov 16 2011 */
%Y A052848 Cf. A000296.
%K A052848 nonn,easy
%O A052848 0,3
%A A052848 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052848 Better name from _Geoffrey Critzer_, Dec 10 2012