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 A291203 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Positive elements in column t=1 give A034855. Elements in rows h=0 give A023531. Elements in rows h=1 give A059297. Positive row sums per layer give A235595. Positive column sums per layer give A061356. LINKS Alois P. Heinz, Layers n = 0..48, flattened FORMULA Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n). 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(2n,n,n) = A126804(n) for n>0. F(n,0,n) = 1 = A000012(n). F(n,1,1) = n = A001477(n) for n>1. F(n,n-1,1) = n! = A000142(n) for n>0. F(n,1,n-1) = A002378(n-1) for n>0. F(n,2,1) = A000551(n). F(n,3,1) = A000552(n). F(n,4,1) = A000553(n). F(n,1,2) = A001788(n-1) for n>2. F(n,0,0) = A000007(n). EXAMPLE n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272 -----+-------------------+---------+------------------+-------- 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 -----+-------------------+---------+------------------+-------- MAPLE 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); CROSSREFS Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529. Sequence in context: A227835 A281154 A245536 * A256852 A128616 A270417 Adjacent sequences:  A291200 A291201 A291202 * A291204 A291205 A291206 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Aug 20 2017 STATUS approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)