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%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
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