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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 (list; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Layers n = 0..48, flattened

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);

CROSSREFS

Cf. A000007, A000041, A000065, A000081, A001853, A005197, A008284, A023531, A033185, A034781, A291203, A291204, A291529.

Sequence in context: A320658 A284966 A143540 * A208664 A030200 A287072

Adjacent sequences:  A291333 A291334 A291335 * A291337 A291338 A291339

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Aug 22 2017

STATUS

approved

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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)