A291204 - OEIS

%I #36 Mar 17 2022 11:38:38

%S 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,

%T 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,

%U 31,90,65,15,0,51,174,120,20,0,54,63,15,0,13,6,0,1,0

%N 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.

%C Elements in rows h=0 give A023531.

%C Positive elements in rows h=1 give A008277.

%C Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A179454.

%C Positive column sums per layer give A132393.

%H Alois P. Heinz, <a href="/A291204/b291204.txt">Layers n = 0..48, flattened</a>

%F Sum_{i=0..n} F(n,i,n-i) = A000325(n).

%F Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000142(n).

%F Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A000254(n).

%F Sum_{t=0..n-1} F(n,1,t) = A058692(n) =  A000110(n) - 1.

%F F(2n,n,n) = A001791(n) for n>0.

%F F(2n,1,n) = A007820(n).

%F F(n,1,n-1) = A000217(n-1) for n>0.

%F F(n,n-1,1) = A057427(n).

%F F(n,1,2) = A000225(n-1) for n>2.

%F F(n,0,n) = 1 = A000012(n).

%F F(n,0,0) = A000007(n).

%e n h\t: 0  1  2  3  4 5 : A179454 : A132393       : A000142

%e -----+-----------------+---------+---------------+--------

%e 0 0  : 1               :       1 :  1            : 1

%e -----+-----------------+---------+---------------+--------

%e 1 0  : 0  1            :       1 :  .            :

%e 1 1  : 0               :         :  1            : 1

%e -----+-----------------+---------+---------------+--------

%e 2 0  : 0  0  1         :       1 :  .  .         :

%e 2 1  : 0  1            :       1 :  .            :

%e 2 2  : 0               :         :  1  1         : 2

%e -----+-----------------+---------+---------------+--------

%e 3 0  : 0  0  0  1      :       1 :  .  .  .      :

%e 3 1  : 0  1  3         :       4 :  .  .         :

%e 3 2  : 0  1            :       1 :  .            :

%e 3 3  : 0               :         :  2  3  1      : 6

%e -----+-----------------+---------+---------------+--------

%e 4 0  : 0  0  0  0  1   :       1 :  .  .  .  .   :

%e 4 1  : 0  1  7  6      :      14 :  .  .  .      :

%e 4 2  : 0  4  4         :       8 :  .  .         :

%e 4 3  : 0  1            :       1 :  .            :

%e 4 4  : 0               :         :  6 11  6  1   : 24

%e -----+-----------------+---------+---------------+--------

%e 5 0  : 0  0  0  0  0 1 :       1 :  .  .  .  . . :

%e 5 1  : 0  1 15 25 10   :      51 :  .  .  .  .   :

%e 5 2  : 0 14 30 10      :      54 :  .  .  .      :

%e 5 3  : 0  8  5         :      13 :  .  .         :

%e 5 4  : 0  1            :       1 :  .            :

%e 5 5  : 0               :         : 24 50 35 10 1 : 120

%e -----+-----------------+---------+---------------+--------

%p b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(

%p        binomial(n-1, j-1)*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))

%p     end:

%p g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):

%p F:= (n, h, t)-> coeff(g(n, h), x, t):

%p seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

%t 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}]]];

%t g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];

%t F[n_, h_, t_] := Coefficient[g[n, h], x, t];

%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_ *)

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

%K nonn,look,tabf

%O 0,17

%A _Alois P. Heinz_, Aug 20 2017