A291336 - OEIS

%I #40 Mar 10 2022 07:53:03

%S 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,

%T 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,

%U 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

%N 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.

%C Elements in rows h=0 give A023531.

%C Positive elements in rows h=1 give A008284.

%C Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A034781.

%C Positive column sums per layer give A033185.

%H Alois P. Heinz, <a href="/A291336/b291336.txt">Layers n = 0..48, flattened</a>

%F Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000081(n+1).

%F Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A005197(n).

%F Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A001853(n+1) for n>0.

%F Sum_{t=0..n-1} F(n,1,t) = A000065(n) = A000041(n) - 1.

%F F(n,1,1) = 1 for n>1.

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

%e n h\t: 0 1 2 3 4 5 : A034781 : A033185   : A000081

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

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

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

%e 3 3  : 0           :         : 2 1 1     : 4

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

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

%e 4 1  : 0 1 2 1     :       4 : . . .     :

%e 4 2  : 0 2 1       :       3 : . .       :

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

%e 4 4  : 0           :         : 4 3 1 1   : 9

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

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

%e 5 1  : 0 1 2 2 1   :       6 : . . . .   :

%e 5 2  : 0 4 3 1     :       8 : . . .     :

%e 5 3  : 0 3 1       :       4 : . .       :

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

%e 5 5  : 0           :         : 9 6 3 1 1 : 20

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

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

%p        or i=1, x^(t*n), b(n, i-1, t, h)+add(x^(t*j)*binomial(

%p        b(i-1$2, 0, h-1)+j-1, j)*b(n-i*j, i-1, t, h), j=1..n/i)))

%p     end:

%p g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 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..9);

%t b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0

%t      || i == 1, x^(t*n), b[n, i-1, t, h] + Sum[x^(t*j)*Binomial[

%t      b[i-1, i-1, 0, h-1]+j-1, j]*b[n - i*j, i-1, t, h], {j, 1, n/i}]]];

%t g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, 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, 9}] //

%t Flatten (* _Jean-François Alcover_, Mar 10 2022, after _Alois P. Heinz_ *)

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

%K nonn,look,tabf

%O 0,28

%A _Alois P. Heinz_, Aug 22 2017