A291203 Number F(n,h,t) of forests of t labeled rooted 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, 1, 0, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0
Offset: 0
Examples
n h\t: 0 1 2 3 4 5 : A235595 : A061356 : A000272 -----+-------------------+---------+------------------+-------- 0 0 : 1 : : : 1 -----+-------------------+---------+------------------+-------- 1 0 : 0 1 : 1 : . : 1 1 : 0 : : 1 : 1 -----+-------------------+---------+------------------+-------- 2 0 : 0 0 1 : 1 : . . : 2 1 : 0 2 : 2 : . : 2 2 : 0 : : 2 1 : 3 -----+-------------------+---------+------------------+-------- 3 0 : 0 0 0 1 : 1 : . . . : 3 1 : 0 3 6 : 9 : . . : 3 2 : 0 6 : 6 : . : 3 3 : 0 : : 9 6 1 : 16 -----+-------------------+---------+------------------+-------- 4 0 : 0 0 0 0 1 : 1 : . . . . : 4 1 : 0 4 24 12 : 40 : . . . : 4 2 : 0 36 24 : 60 : . . : 4 3 : 0 24 : 24 : . : 4 4 : 0 : : 64 48 12 1 : 125 -----+-------------------+---------+------------------+-------- 5 0 : 0 0 0 0 0 1 : 1 : . . . . . : 5 1 : 0 5 80 90 20 : 195 : . . . . : 5 2 : 0 200 300 60 : 560 : . . . : 5 3 : 0 300 120 : 420 : . . : 5 4 : 0 120 : 120 : . : 5 5 : 0 : : 625 500 150 20 1 : 1296 -----+-------------------+---------+------------------+--------
Links
- Alois P. Heinz, Layers n = 0..48, flattened
Crossrefs
Programs
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Maple
b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add( binomial(n-1, j-1)*j*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]*j*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 *)
Formula
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
F(n,0,0) = A000007(n).
Comments