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%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
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