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%I #39 Mar 17 2022 11:42:06
%S 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,
%T 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,
%U 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
%N 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.
%C Positive elements in column t=1 give A034855.
%C Elements in rows h=0 give A023531.
%C Elements in rows h=1 give A059297.
%C Positive row sums per layer give A235595.
%C Positive column sums per layer give A061356.
%H Alois P. Heinz, <a href="/A291203/b291203.txt">Layers n = 0..48, flattened</a>
%F Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
%F Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
%F Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
%F Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
%F Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
%F Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
%F F(2n,n,n) = A126804(n) for n>0.
%F F(n,0,n) = 1 = A000012(n).
%F F(n,1,1) = n = A001477(n) for n>1.
%F F(n,n-1,1) = n! = A000142(n) for n>0.
%F F(n,1,n-1) = A002378(n-1) for n>0.
%F F(n,2,1) = A000551(n).
%F F(n,3,1) = A000552(n).
%F F(n,4,1) = A000553(n).
%F F(n,1,2) = A001788(n-1) for n>2.
%F F(n,0,0) = A000007(n).
%e n h\t: 0 1 2 3 4 5 : A235595 : A061356 : A000272
%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 2 : 2 : . :
%e 2 2 : 0 : : 2 1 : 3
%e -----+-------------------+---------+------------------+--------
%e 3 0 : 0 0 0 1 : 1 : . . . :
%e 3 1 : 0 3 6 : 9 : . . :
%e 3 2 : 0 6 : 6 : . :
%e 3 3 : 0 : : 9 6 1 : 16
%e -----+-------------------+---------+------------------+--------
%e 4 0 : 0 0 0 0 1 : 1 : . . . . :
%e 4 1 : 0 4 24 12 : 40 : . . . :
%e 4 2 : 0 36 24 : 60 : . . :
%e 4 3 : 0 24 : 24 : . :
%e 4 4 : 0 : : 64 48 12 1 : 125
%e -----+-------------------+---------+------------------+--------
%e 5 0 : 0 0 0 0 0 1 : 1 : . . . . . :
%e 5 1 : 0 5 80 90 20 : 195 : . . . . :
%e 5 2 : 0 200 300 60 : 560 : . . . :
%e 5 3 : 0 300 120 : 420 : . . :
%e 5 4 : 0 120 : 120 : . :
%e 5 5 : 0 : : 625 500 150 20 1 : 1296
%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)*j*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[
%t Binomial[n-1, j-1]*j*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, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.
%K nonn,look,tabf
%O 0,9
%A _Alois P. Heinz_, Aug 20 2017
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