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
Examples
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 -----+-------------+---------+-----------+--------
Links
- Alois P. Heinz, Layers n = 0..48, flattened
Programs
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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 *)
Comments