A291529 Number F(n,h,t) of forests of t (unlabeled) rooted identity trees with n vertices such that 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, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 5, 1, 0, 0, 5, 4, 0, 0, 4, 1, 0, 1, 0

Offset

0,49

Comments

Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A227819.

Positive column sums per layer give A227774.

Links

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

Formula

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A004111(n+1).

Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A291532(n).

Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * F(n-1,h,t) = A291559(n).

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

Example

n h\t: 0 1 2 3 4 5 : A227819 : A227774   : A004111
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 0       :       0 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 0       : 1
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 0     :       0 : . . .     :
3 1  : 0 0 1       :       1 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 1 1 0     : 2
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 0   :       0 : . . . .   :
4 1  : 0 0 0 0     :       0 : . . .     :
4 2  : 0 1 1       :       2 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 2 1 0 0   : 3
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 0 :       0 : . . . . . :
5 1  : 0 0 0 0 0   :       0 : . . . .   :
5 2  : 0 0 2 0     :       2 : . . .     :
5 3  : 0 2 1       :       3 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 3 3 0 0 0 : 6
-----+-------------+---------+-----------+--------

Maple

b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
      `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
       b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
    end:
g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 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..10);

Mathematica

b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1, If[n < 2, x^(t*n), 0], b[n, i - 1, t, h] + Sum[x^(t*j)*Binomial[b[i - 1, i - 1, 0, h - 1], j]*b[n - i*j, i - 1, t, h], {j, 1, n/i}]]];
g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[F[n, h, t], {n, 0, 10}, {h, 0, n}, {t, 0, n - h}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Crossrefs

Cf. A000007, A004111, A227774, A227819, A291203, A291204, A291336, A291532, A291559.

Sequence in context: A353836 A161364 A143620 * A366981 A236417 A238304

Adjacent sequences: A291526 A291527 A291528 * A291530 A291531 A291532

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Aug 25 2017

Status

approved