OFFSET
0,4
COMMENTS
A functional digraph of a function f:{1,2,...,n}->{1,2,...,n} is a directed graph on vertex set {1,2,...,n} with an arrow from i to j if f(i)=j. Every connected component of the digraph contains a unique cycle and every vertex i of this cycle is the root of a rooted tree directed towards i. T(n,k) is the number k of rooted trees that consist of a single vertex over all cycles in all functional digraphs on {1,2,...,n}. Definition from Stanley, page 83.
Row sums = n^n
REFERENCES
R. Stanley, Enumerative Combinatorics Vol II, Cambridge Univ. Press, 1999.
FORMULA
E.g.f.:1/(1 - x*(exp(T(x) -1 +y)) where T(x) is the e.g.f. for A000169.
EXAMPLE
0,
0, 1,
2, 0, 2,
9, 12, 0, 6,
88, 72, 72, 0, 24,
985, 1000, 540, 480, 0, 120,
13956, 13980, 10080, 4320, 3600, 0, 720
MATHEMATICA
nn=6; f[list_]:=Select[list, #>0&]; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Prepend[Drop[Map[Insert[#, 0, -2]&, Map[f, Range[0, nn]!CoefficientList[Series[1/(1-x(Exp[t]-1+y)), {x, 0, nn}], {x, y}]]], 1], {0}]//Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 08 2012
STATUS
approved