A219034 Triangular array read by rows: T(n,k) is the number of forests of rooted trees on n labeled nodes with exactly k isolated nodes; n>=0, 0<=k<=n.

1, 0, 1, 2, 0, 1, 9, 6, 0, 1, 76, 36, 12, 0, 1, 805, 380, 90, 20, 0, 1, 10626, 4830, 1140, 180, 30, 0, 1, 167839, 74382, 16905, 2660, 315, 42, 0, 1, 3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1, 65127465, 27825912, 6042204, 892584, 101430, 9576, 756, 72, 0, 1

Offset: 0

Author

Geoffrey Critzer, Nov 10 2012

Keywords

• nonn
• tabl

Comments

Column k=0 is A105785.

Row sums = (n+1)^(n-1).

Examples

			Triangle T(n,k) begins:
        1;
        0,       1;
        2,       0,      1;
        9,       6,      0,     1;
       76,      36,     12,     0,    1;
      805,     380,     90,    20,    0,   1;
    10626,    4830,   1140,   180,   30,   0,  1;
   167839,   74382,  16905,  2660,  315,  42,  0, 1;
  3091768, 1342712, 297528, 45080, 5320, 504, 56, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Programs

Maple

b:= proc(n) option remember; expand(`if`(n=0, 1, add(i^(i-1)
      *b(n-i)*binomial(n-1, i-1)*`if`(i=1, x, 1), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 31 2021

Mathematica

nn=8; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Range[0,nn]! CoefficientList[ Series[Exp[t-x+y x], {x,0,nn}], {x,y}] //Grid

Formula

E.g.f.: exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.