A228534 Triangular array read by rows: T(n,k) is the number of functional digraphs on {1,2,...,n} such that every element is mapped to a recurrent element and there are exactly k cycles, n>=1, 1<=k<=n.

1, 3, 1, 11, 9, 1, 58, 71, 18, 1, 409, 620, 245, 30, 1, 3606, 6274, 3255, 625, 45, 1, 38149, 73339, 45724, 11795, 1330, 63, 1, 470856, 977780, 697004, 221529, 33880, 2506, 84, 1, 6641793, 14678712, 11602394, 4309956, 823179, 82908, 4326, 108, 1

Offset: 1

Author

Geoffrey Critzer, Aug 24 2013

Keywords

• nonn
• tabl

Comments

The Bell transform of (-1)^n*A009444(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			       1;
       3,      1;
      11,      9,      1;
      58,     71,     18,      1;
     409,    620,    245,     30,     1;
    3606,   6274,   3255,    625,    45,    1;
   38149,  73339,  45724,  11795,  1330,   63,  1;
  470856, 977780, 697004, 221529, 33880, 2506, 84, 1;
		

Links

• Alois P. Heinz, Rows n = 1..90, flattened

Crossrefs

Row sums = A006153.

Column 1 = |A009444|.

Cf. A199673.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
g := n -> add(m^(n-m)*m!*binomial(n+1,m), m=1..n+1);
BellMatrix(g, 9); # Peter Luschny, Jan 29 2016

Mathematica

nn = 8; a = x Exp[x];
Map[Select[#, # > 0 &] &,
  Drop[Range[0, nn]! CoefficientList[
     Series[1/(1 - a)^y, {x, 0, nn}], {x, y}], 1]] // Grid
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (n+1)! Sum[m^(n-m)/(n-m+1)!, {m, 1, n+1}]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Sage

# uses[bell_matrix from A264428, A009444]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n*A009444(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

E.g.f.: 1/(1 - x*exp(x))^y.