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 A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows. 16
 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 2, 5, 4, 1, 0, 0, 0, 2, 8, 9, 5, 1, 0, 0, 0, 1, 12, 18, 14, 6, 1, 0, 0, 0, 1, 17, 34, 33, 20, 7, 1, 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1, 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1, 0, 0, 0, 0, 41, 187, 301, 303, 221, 118, 44, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,14 LINKS Alois P. Heinz, Rows n = 1..141, flattened EXAMPLE :   T(6,4) = 3              :  T(11,3) = 1  : :     o       o       o     :        o      : :    / \      |       |     :      /( )\    : :   o   o     o       o     :     o o o o   : :   |        / \      |     :    /| | |     : :   o       o   o     o     :   o o o o     : :   |       |        / \    :   |   |       : :   o       o       o   o   :   o   o       : :   |       |       |       :               : :   o       o       o       :               : Triangle T(n,k) begins:   1;   0, 1;   0, 0, 1;   0, 0, 1, 1;   0, 0, 0, 2,  1;   0, 0, 0, 2,  3,   1;   0, 0, 0, 2,  5,   4,   1;   0, 0, 0, 2,  8,   9,   5,   1;   0, 0, 0, 1, 12,  18,  14,   6,  1;   0, 0, 0, 1, 17,  34,  33,  20,  7,  1;   0, 0, 0, 1, 23,  61,  72,  54, 27,  8, 1;   0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1; MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,       add(binomial(b((i-1)\$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))     end: T:= (n, k)-> b((n-1)\$2, k) -`if`(k=0, 0, b((n-1)\$2, k-1)): seq(seq(T(n, k), k=0..n-1), n=1..15); MATHEMATICA Drop[Transpose[Map[PadRight[#, 15]&, Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 15}], x]&, {1}, n]; f[m]-PadRight[f[m-1], Length[f[m]]], {m, 1, 15}]]], 1]//Grid (* Geoffrey Critzer, Aug 01 2013 *) CROSSREFS Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404. Row sums give: A004111. Column sums give: A038081. Largest n with T(n,k)>0 is A038093(k). Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830. T(2n,n) gives A245090. T(2n+1,n) gives A245091. Cf. A034781. Sequence in context: A093323 A106278 A177517 * A064287 A196389 A128206 Adjacent sequences:  A227816 A227817 A227818 * A227820 A227821 A227822 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 31 2013 STATUS approved

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)