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

0,17

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.

Links

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

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).

Example

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
-----+-----------------+---------+---------------+--------

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 *)

Crossrefs

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

Sequence in context: A009138 A175562 A319330 * A331414 A111025 A271620

Adjacent sequences: A291201 A291202 A291203 * A291205 A291206 A291207

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Aug 20 2017

Status

approved