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A291204
Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that the root of each subtree contains the subtree's minimal label and 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.
4
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 7, 6, 0, 4, 4, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 15, 25, 10, 0, 14, 30, 10, 0, 8, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 31, 90, 65, 15, 0, 51, 174, 120, 20, 0, 54, 63, 15, 0, 13, 6, 0, 1, 0
OFFSET
0,17
COMMENTS
Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008277.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A179454.
Positive column sums per layer give A132393.
LINKS
FORMULA
Sum_{i=0..n} F(n,i,n-i) = A000325(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000142(n).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A000254(n).
Sum_{t=0..n-1} F(n,1,t) = A058692(n) = A000110(n) - 1.
F(2n,n,n) = A001791(n) for n>0.
F(2n,1,n) = A007820(n).
F(n,1,n-1) = A000217(n-1) for n>0.
F(n,n-1,1) = A057427(n).
F(n,1,2) = A000225(n-1) for n>2.
F(n,0,n) = 1 = A000012(n).
F(n,0,0) = A000007(n).
EXAMPLE
n h\t: 0 1 2 3 4 5 : A179454 : A132393 : A000142
-----+-----------------+---------+---------------+--------
0 0 : 1 : 1 : 1 : 1
-----+-----------------+---------+---------------+--------
1 0 : 0 1 : 1 : . :
1 1 : 0 : : 1 : 1
-----+-----------------+---------+---------------+--------
2 0 : 0 0 1 : 1 : . . :
2 1 : 0 1 : 1 : . :
2 2 : 0 : : 1 1 : 2
-----+-----------------+---------+---------------+--------
3 0 : 0 0 0 1 : 1 : . . . :
3 1 : 0 1 3 : 4 : . . :
3 2 : 0 1 : 1 : . :
3 3 : 0 : : 2 3 1 : 6
-----+-----------------+---------+---------------+--------
4 0 : 0 0 0 0 1 : 1 : . . . . :
4 1 : 0 1 7 6 : 14 : . . . :
4 2 : 0 4 4 : 8 : . . :
4 3 : 0 1 : 1 : . :
4 4 : 0 : : 6 11 6 1 : 24
-----+-----------------+---------+---------------+--------
5 0 : 0 0 0 0 0 1 : 1 : . . . . . :
5 1 : 0 1 15 25 10 : 51 : . . . . :
5 2 : 0 14 30 10 : 54 : . . . :
5 3 : 0 8 5 : 13 : . . :
5 4 : 0 1 : 1 : . :
5 5 : 0 : : 24 50 35 10 1 : 120
-----+-----------------+---------+---------------+--------
MAPLE
b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
binomial(n-1, j-1)*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]*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 *)
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Aug 20 2017
STATUS
approved