A291204 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that the root of each subtree contains the subtree's minimal label and h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 7, 6, 0, 4, 4, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 15, 25, 10, 0, 14, 30, 10, 0, 8, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 31, 90, 65, 15, 0, 51, 174, 120, 20, 0, 54, 63, 15, 0, 13, 6, 0, 1, 0

Offset: 0

Author

Alois P. Heinz, Aug 20 2017

Keywords

• nonn
• look
• tabf

Comments

Elements in rows h=0 give A023531.

Positive elements in rows h=1 give A008277.

Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A179454.

Positive column sums per layer give A132393.

Examples

			n h\t: 0  1  2  3  4 5 : A179454 : A132393       : A000142
-----+-----------------+---------+---------------+--------
0 0  : 1               :       1 :  1            : 1
-----+-----------------+---------+---------------+--------
1 0  : 0  1            :       1 :  .            :
1 1  : 0               :         :  1            : 1
-----+-----------------+---------+---------------+--------
2 0  : 0  0  1         :       1 :  .  .         :
2 1  : 0  1            :       1 :  .            :
2 2  : 0               :         :  1  1         : 2
-----+-----------------+---------+---------------+--------
3 0  : 0  0  0  1      :       1 :  .  .  .      :
3 1  : 0  1  3         :       4 :  .  .         :
3 2  : 0  1            :       1 :  .            :
3 3  : 0               :         :  2  3  1      : 6
-----+-----------------+---------+---------------+--------
4 0  : 0  0  0  0  1   :       1 :  .  .  .  .   :
4 1  : 0  1  7  6      :      14 :  .  .  .      :
4 2  : 0  4  4         :       8 :  .  .         :
4 3  : 0  1            :       1 :  .            :
4 4  : 0               :         :  6 11  6  1   : 24
-----+-----------------+---------+---------------+--------
5 0  : 0  0  0  0  0 1 :       1 :  .  .  .  . . :
5 1  : 0  1 15 25 10   :      51 :  .  .  .  .   :
5 2  : 0 14 30 10      :      54 :  .  .  .      :
5 3  : 0  8  5         :      13 :  .  .         :
5 4  : 0  1            :       1 :  .            :
5 5  : 0               :         : 24 50 35 10 1 : 120
-----+-----------------+---------+---------------+--------
		

Links

• Alois P. Heinz, Layers n = 0..48, flattened

Crossrefs

Cf. A000007, A000012, A000110, A000142, A000217, A000225, A000254, A000325, A001791, A007820, A008277, A023531, A048993, A057427, A058692, A179454, A291203, A291336, A291529.

Programs

Maple

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

Mathematica

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[Binomial[n-1, j-1]*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Formula

Sum_{i=0..n} F(n,i,n-i) = A000325(n).

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000142(n).

Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A000254(n).

Sum_{t=0..n-1} F(n,1,t) = A058692(n) = A000110(n) - 1.

F(2n,n,n) = A001791(n) for n>0.

F(2n,1,n) = A007820(n).

F(n,1,n-1) = A000217(n-1) for n>0.

F(n,n-1,1) = A057427(n).

F(n,1,2) = A000225(n-1) for n>2.

F(n,0,n) = 1 = A000012(n).

F(n,0,0) = A000007(n).