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A291203
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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.
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4
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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, 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, 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
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OFFSET
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0,9
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COMMENTS
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Positive elements in column t=1 give A034855.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.
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LINKS
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FORMULA
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Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(n,n-1,1) = n! = A000142(n) for n>0.
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EXAMPLE
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-----+-------------------+---------+------------------+--------
0 0 : 1 : : : 1
-----+-------------------+---------+------------------+--------
1 0 : 0 1 : 1 : . :
1 1 : 0 : : 1 : 1
-----+-------------------+---------+------------------+--------
2 0 : 0 0 1 : 1 : . . :
2 1 : 0 2 : 2 : . :
2 2 : 0 : : 2 1 : 3
-----+-------------------+---------+------------------+--------
3 0 : 0 0 0 1 : 1 : . . . :
3 1 : 0 3 6 : 9 : . . :
3 2 : 0 6 : 6 : . :
3 3 : 0 : : 9 6 1 : 16
-----+-------------------+---------+------------------+--------
4 0 : 0 0 0 0 1 : 1 : . . . . :
4 1 : 0 4 24 12 : 40 : . . . :
4 2 : 0 36 24 : 60 : . . :
4 3 : 0 24 : 24 : . :
4 4 : 0 : : 64 48 12 1 : 125
-----+-------------------+---------+------------------+--------
5 0 : 0 0 0 0 0 1 : 1 : . . . . . :
5 1 : 0 5 80 90 20 : 195 : . . . . :
5 2 : 0 200 300 60 : 560 : . . . :
5 3 : 0 300 120 : 420 : . . :
5 4 : 0 120 : 120 : . :
5 5 : 0 : : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------
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MAPLE
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b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
binomial(n-1, j-1)*j*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);
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MATHEMATICA
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b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
Binomial[n-1, j-1]*j*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, 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 *)
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CROSSREFS
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Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.
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KEYWORD
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AUTHOR
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STATUS
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approved
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