%I #16 Jun 04 2018 08:20:56

%S 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,

%T 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,

%U 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

%N 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.

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

%C Positive column sums per layer give A227774.

%H Alois P. Heinz, <a href="/A291529/b291529.txt">Layers n = 0..48, flattened</a>

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

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

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

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

%e n h\t: 0 1 2 3 4 5 : A227819 : A227774 : A004111

%e -----+-------------+---------+-----------+--------

%e 0 0 : 1 : : : 1

%e -----+-------------+---------+-----------+--------

%e 1 0 : 0 1 : 1 : . :

%e 1 1 : 0 : : 1 : 1

%e -----+-------------+---------+-----------+--------

%e 2 0 : 0 0 0 : 0 : . . :

%e 2 1 : 0 1 : 1 : . :

%e 2 2 : 0 : : 1 0 : 1

%e -----+-------------+---------+-----------+--------

%e 3 0 : 0 0 0 0 : 0 : . . . :

%e 3 1 : 0 0 1 : 1 : . . :

%e 3 2 : 0 1 : 1 : . :

%e 3 3 : 0 : : 1 1 0 : 2

%e -----+-------------+---------+-----------+--------

%e 4 0 : 0 0 0 0 0 : 0 : . . . . :

%e 4 1 : 0 0 0 0 : 0 : . . . :

%e 4 2 : 0 1 1 : 2 : . . :

%e 4 3 : 0 1 : 1 : . :

%e 4 4 : 0 : : 2 1 0 0 : 3

%e -----+-------------+---------+-----------+--------

%e 5 0 : 0 0 0 0 0 0 : 0 : . . . . . :

%e 5 1 : 0 0 0 0 0 : 0 : . . . . :

%e 5 2 : 0 0 2 0 : 2 : . . . :

%e 5 3 : 0 2 1 : 3 : . . :

%e 5 4 : 0 1 : 1 : . :

%e 5 5 : 0 : : 3 3 0 0 0 : 6

%e -----+-------------+---------+-----------+--------

%p b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,

%p `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(

%p b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))

%p end:

%p g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):

%p F:= (n, h, t)-> coeff(g(n, h), x, t):

%p seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..10);

%t 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}]]];

%t g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h - 1]];

%t F[n_, h_, t_] := Coefficient[g[n, h], x, t];

%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 *)

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

%K nonn,look,tabf

%O 0,49

%A _Alois P. Heinz_, Aug 25 2017