login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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

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

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);

CROSSREFS

Cf. A000007, A000012, A000110, A000142, A000217, A000225, A000254, A000325, A001791, A007820, A008277, A023531, A048993, A057427, A058692, A179454, A291203, A291336, A291529.

Sequence in context: A009138 A175562 A319330 * A331414 A111025 A271620

Adjacent sequences:  A291201 A291202 A291203 * A291205 A291206 A291207

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Aug 20 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)