|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,17
|
|
COMMENTS
|
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).
|
|
EXAMPLE
|
-----+-----------------+---------+---------------+--------
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 *)
|
|
CROSSREFS
|
Cf. A000007, A000012, A000110, A000142, A000217, A000225, A000254, A000325, A001791, A007820, A008277, A023531, A048993, A057427, A058692, A179454, A291203, A291336, A291529.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|