A144258 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows: T(n,k) is the number of forests of trees on n or fewer nodes using a subset of labels 1..n and k edges.

1, 2, 0, 4, 1, 0, 8, 6, 3, 0, 16, 24, 27, 16, 0, 32, 80, 150, 190, 125, 0, 64, 240, 660, 1335, 1830, 1296, 0, 128, 672, 2520, 7210, 15435, 22449, 16807, 0, 256, 1792, 8736, 33040, 98105, 219912, 335160, 262144, 0, 512, 4608, 28224, 135072, 521010, 1600452, 3727962, 5902236, 4782969, 0

Offset: 0

Author

Alois P. Heinz, Sep 16 2008

Keywords

• nonn
• tabl

Examples

			T(3,1) = 6, because there are 6 forests of trees on 3 or fewer nodes using a subset of labels 1,2,3 and 1 edge:
  .1-2. .1... ...2. .1-2. .1.2. .1.2.
  ..... .|... ../.. ..... .|... ../..
  ..... .3... .3... .3... .3... .3...
Triangle begins:
   1;
   2,  0;
   4,  1,   0;
   8,  6,   3,   0;
  16, 24,  27,  16,   0;
  32, 80, 150, 190, 125,  0;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Index entries for sequences related to trees

Crossrefs

Columns k = 0, 1 give A000079, A001788.

First lower diagonal gives A000272(k+1) with initial term 2.

Row sums give A144259.

Cf. A007318, A000142.

Programs

Maple

T:= proc(n, k) option remember;
      if k=0 then 2^n
    elif k<0 or n<=k then 0
    elif k=n-1 then n^(n-2)
    else add(binomial(n-1, j) *T(j+1, j) *T(n-1-j, k-j), j=0..k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..11);

Mathematica

T[n_, k_] := T[n, k] = Which[k == 0, 2^n, k < 0 || n <= k, 0, k == n-1, n^(n-2), True, Sum[Binomial[n-1, j]*T[j+1, j]*T[n-1-j, k-j], {j, 0, k}]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2014, translated from Maple *)

Formula

T(n,0) = 2^n, T(n,k) = 0 if k < 0 or n <= k, otherwise T(n,k) = n^(n-2) if k=n-1, otherwise T(n,k) = Sum_{j=0..k} C(n-1,j)*T(j+1,j)*T(n-1-j,k-j).