A143911 Triangle read by rows: T(n,k) = number of forests on n labeled nodes, where k is the maximum of the number of edges per tree (n>=1, 0<=k<=n-1).

1, 1, 1, 1, 3, 3, 1, 9, 12, 16, 1, 25, 60, 80, 125, 1, 75, 330, 480, 750, 1296, 1, 231, 1680, 3920, 5250, 9072, 16807, 1, 763, 9408, 33600, 49000, 72576, 134456, 262144, 1, 2619, 56952, 254016, 598500, 762048, 1210104, 2359296, 4782969, 1, 9495, 348120

Offset: 1

Author

Alois P. Heinz, Sep 04 2008

Keywords

• nonn
• tabl

Examples

			T(4,1) = 9, because 9 forests on 4 labeled nodes have 1 as the maximum of the number of edges per tree:
  .1-2. .1.2. .1.2. .1.2. .1.2. .1.2. .1-2. .1.2. .1.2.
  ..... ...|. ..... .|... ..\.. ../.. ..... .|.|. ..X..
  .4.3. .4.3. .4-3. .4.3. .4.3. .4.3. .4-3. .4.3. .4.3.
Triangle begins:
  1;
  1,  1;
  1,  3,   3;
  1,  9,  12,  16;
  1, 25,  60,  80, 125;
  1, 75, 330, 480, 750, 1296;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Columns k=0-1 give: A000012, A001189.

Row sums give A001858.

Rightmost diagonal gives A000272.

Cf. A138464.

Programs

Maple

A:= (n,k)-> coeff(series(exp(add(j^(j-2) *x^j/j!, j=1..k)), x, n+1), x,n)*n!: T:= (n,k)-> A(n,k+1)-A(n,k): seq(seq(T(n,k), k=0..n-1), n=1..11);

Mathematica

A[n_, k_] := SeriesCoefficient[Exp[Sum[j^(j-2)*x^j/j!, {j, 1, k}]], {x, 0, n}]*n!; T[n_, k_] := A[n, k+1] - A[n, k];
Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, May 31 2016, translated from Maple *)

Formula

See program.