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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.

%I A291529 #16 Jun 04 2018 08:20:56
%S A291529 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 A291529 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 A291529 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 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.
%C A291529 Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A227819.
%C A291529 Positive column sums per layer give A227774.
%H A291529 Alois P. Heinz, <a href="/A291529/b291529.txt">Layers n = 0..48, flattened</a>
%F A291529 Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A004111(n+1).
%F A291529 Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A291532(n).
%F A291529 Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * F(n-1,h,t) = A291559(n).
%F A291529 F(n,0,0) = A000007(n).
%e A291529 n h\t: 0 1 2 3 4 5 : A227819 : A227774   : A004111
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 0 0  : 1           :         :           : 1
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 1 0  : 0 1         :       1 : .         :
%e A291529 1 1  : 0           :         : 1         : 1
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 2 0  : 0 0 0       :       0 : . .       :
%e A291529 2 1  : 0 1         :       1 : .         :
%e A291529 2 2  : 0           :         : 1 0       : 1
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 3 0  : 0 0 0 0     :       0 : . . .     :
%e A291529 3 1  : 0 0 1       :       1 : . .       :
%e A291529 3 2  : 0 1         :       1 : .         :
%e A291529 3 3  : 0           :         : 1 1 0     : 2
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 4 0  : 0 0 0 0 0   :       0 : . . . .   :
%e A291529 4 1  : 0 0 0 0     :       0 : . . .     :
%e A291529 4 2  : 0 1 1       :       2 : . .       :
%e A291529 4 3  : 0 1         :       1 : .         :
%e A291529 4 4  : 0           :         : 2 1 0 0   : 3
%e A291529 -----+-------------+---------+-----------+--------
%e A291529 5 0  : 0 0 0 0 0 0 :       0 : . . . . . :
%e A291529 5 1  : 0 0 0 0 0   :       0 : . . . .   :
%e A291529 5 2  : 0 0 2 0     :       2 : . . .     :
%e A291529 5 3  : 0 2 1       :       3 : . .       :
%e A291529 5 4  : 0 1         :       1 : .         :
%e A291529 5 5  : 0           :         : 3 3 0 0 0 : 6
%e A291529 -----+-------------+---------+-----------+--------
%p A291529 b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
%p A291529       `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
%p A291529        b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
%p A291529     end:
%p A291529 g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
%p A291529 F:= (n, h, t)-> coeff(g(n, h), x, t):
%p A291529 seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..10);
%t A291529 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 A291529 g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h - 1]];
%t A291529 F[n_, h_, t_] := Coefficient[g[n, h], x, t];
%t A291529 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 A291529 Cf. A000007, A004111, A227774, A227819, A291203, A291204, A291336, A291532, A291559.
%K A291529 nonn,look,tabf
%O A291529 0,49
%A A291529 _Alois P. Heinz_, Aug 25 2017