A144289 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows: Number T(n,k) of forests of labeled rooted trees on n or fewer nodes using a subset of labels 1..n and k edges.

1, 2, 0, 4, 2, 0, 8, 12, 9, 0, 16, 48, 84, 64, 0, 32, 160, 480, 820, 625, 0, 64, 480, 2160, 6120, 10230, 7776, 0, 128, 1344, 8400, 34720, 94500, 155274, 117649, 0, 256, 3584, 29568, 165760, 647920, 1712592, 2776200, 2097152, 0, 512, 9216, 96768, 701568, 3669120, 13783392, 35630784, 57138120, 43046721, 0

Offset: 0

Author

Alois P. Heinz, Sep 17 2008

Keywords

• nonn
• tabl

Examples

			T(3,1) = 12, because there are 12 forests of labeled rooted trees on 3 or fewer nodes using a subset of labels 1..3 and 1 edge:
  .1<2. .2<1. .1<3. .3<1. .2<3. .3<2. .1<2. .2<1. .1<3. .3<1. .2<3. .3<2.
  ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....
  ..... ..... ..... ..... ..... ..... .3... .3... .2... .2... .1... .1...
Triangle begins:
   1;
   2,   0;
   4,   2,   0;
   8,  12,   9,   0;
  16,  48,  84,  64,   0;
  32, 160, 480, 820, 625,  0;
		

Links

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

Crossrefs

Columns 0, 1 give A000079, A001815.

First lower diagonal gives A000169 with first term 2.

Row sums give A088957.

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-1)
    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-1), True, Sum[Binomial[n-1, j]*T[j+1, j]*T[n-1-j, k-j], {j, 0, k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Jan 21 2014, translated from Alois P. Heinz's Maple code *)

Formula

T(n,0) = 2^n, T(n,k) = 0 if k < 0 or n <= k, otherwise T(n,k) = n^(n-1) 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).