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 A291204 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. 4
 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, 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, 31, 90, 65, 15, 0, 51, 174, 120, 20, 0, 54, 63, 15, 0, 13, 6, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,17 COMMENTS Elements in rows h=0 give A023531. Positive elements in rows h=1 give A008277. Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A179454. Positive column sums per layer give A132393. LINKS Alois P. Heinz, Layers n = 0..48, flattened FORMULA Sum_{i=0..n} F(n,i,n-i) = A000325(n). Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000142(n). Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A000254(n). Sum_{t=0..n-1} F(n,1,t) = A058692(n) =  A000110(n) - 1. F(2n,n,n) = A001791(n) for n>0. F(2n,1,n) = A007820(n). F(n,1,n-1) = A000217(n-1) for n>0. F(n,n-1,1) = A057427(n). F(n,1,2) = A000225(n-1) for n>2. F(n,0,n) = 1 = A000012(n). F(n,0,0) = A000007(n). EXAMPLE n h\t: 0  1  2  3  4 5 : A179454 : A132393       : A000142 -----+-----------------+---------+---------------+-------- 0 0  : 1               :       1 :  1            : 1 -----+-----------------+---------+---------------+-------- 1 0  : 0  1            :       1 :  .            : 1 1  : 0               :         :  1            : 1 -----+-----------------+---------+---------------+-------- 2 0  : 0  0  1         :       1 :  .  .         : 2 1  : 0  1            :       1 :  .            : 2 2  : 0               :         :  1  1         : 2 -----+-----------------+---------+---------------+-------- 3 0  : 0  0  0  1      :       1 :  .  .  .      : 3 1  : 0  1  3         :       4 :  .  .         : 3 2  : 0  1            :       1 :  .            : 3 3  : 0               :         :  2  3  1      : 6 -----+-----------------+---------+---------------+-------- 4 0  : 0  0  0  0  1   :       1 :  .  .  .  .   : 4 1  : 0  1  7  6      :      14 :  .  .  .      : 4 2  : 0  4  4         :       8 :  .  .         : 4 3  : 0  1            :       1 :  .            : 4 4  : 0               :         :  6 11  6  1   : 24 -----+-----------------+---------+---------------+-------- 5 0  : 0  0  0  0  0 1 :       1 :  .  .  .  . . : 5 1  : 0  1 15 25 10   :      51 :  .  .  .  .   : 5 2  : 0 14 30 10      :      54 :  .  .  .      : 5 3  : 0  8  5         :      13 :  .  .         : 5 4  : 0  1            :       1 :  .            : 5 5  : 0               :         : 24 50 35 10 1 : 120 -----+-----------------+---------+---------------+-------- MAPLE b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(        binomial(n-1, j-1)*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); MATHEMATICA 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}]]]; 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 *) CROSSREFS Cf. A000007, A000012, A000110, A000142, A000217, A000225, A000254, A000325, A001791, A007820, A008277, A023531, A048993, A057427, A058692, A179454, A291203, A291336, A291529. Sequence in context: A009138 A175562 A319330 * A331414 A111025 A271620 Adjacent sequences:  A291201 A291202 A291203 * A291205 A291206 A291207 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Aug 20 2017 STATUS approved

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Last modified May 20 17:47 EDT 2022. Contains 353876 sequences. (Running on oeis4.)