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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Positive elements in column t=1 give A034855.

Elements in rows h=0 give A023531.

Elements in rows h=1 give A059297.

Positive row sums per layer give A235595.

Positive column sums per layer give A061356.

LINKS

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

FORMULA

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).

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).

EXAMPLE

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

-----+-------------------+---------+------------------+--------

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);

CROSSREFS

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

Sequence in context: A227835 A281154 A245536 * A256852 A128616 A270417

Adjacent sequences:  A291200 A291201 A291202 * A291204 A291205 A291206

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Aug 20 2017

STATUS

approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)