VOOZH about

URL: https://oeis.org/A291336

⇱ A291336 - OEIS

login
A291336
Number F(n,h,t) of forests of t unlabeled 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.
4
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 4, 3, 1, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 3, 2, 1, 0, 6, 8, 3, 1, 0, 8, 4, 1, 0, 4, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 4, 3, 2, 1, 0, 10, 15, 9, 3, 1, 0, 18, 13, 4, 1, 0, 13, 5, 1, 0, 5, 1, 0, 1, 0
OFFSET
0,28
COMMENTS
Elements in rows h=0 give A023531.
Positive elements in rows h=1 give A008284.
Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A034781.
Positive column sums per layer give A033185.
LINKS
FORMULA
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000081(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A005197(n).
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A001853(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A000065(n) = A000041(n) - 1.
F(n,1,1) = 1 for n>1.
F(n,0,0) = A000007(n).
EXAMPLE
n h\t: 0 1 2 3 4 5 : A034781 : A033185 : A000081
-----+-------------+---------+-----------+--------
0 0 : 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 1 : 2 : . . :
3 2 : 0 1 : 1 : . :
3 3 : 0 : : 2 1 1 : 4
-----+-------------+---------+-----------+--------
4 0 : 0 0 0 0 1 : 1 : . . . . :
4 1 : 0 1 2 1 : 4 : . . . :
4 2 : 0 2 1 : 3 : . . :
4 3 : 0 1 : 1 : . :
4 4 : 0 : : 4 3 1 1 : 9
-----+-------------+---------+-----------+--------
5 0 : 0 0 0 0 0 1 : 1 : . . . . . :
5 1 : 0 1 2 2 1 : 6 : . . . . :
5 2 : 0 4 3 1 : 8 : . . . :
5 3 : 0 3 1 : 4 : . . :
5 4 : 0 1 : 1 : . :
5 5 : 0 : : 9 6 3 1 1 : 20
-----+-------------+---------+-----------+--------
MAPLE
b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0
or i=1, x^(t*n), b(n, i-1, t, h)+add(x^(t*j)*binomial(
b(i-1$2, 0, h-1)+j-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..9);
MATHEMATICA
b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0
|| i == 1, x^(t*n), b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[
b[i-1, i-1, 0, h-1]+j-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[Table[Table[F[n, h, t], {t, 0, n-h}], {h, 0, n}], {n, 0, 9}] //
Flatten (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Aug 22 2017
STATUS
approved