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.

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

Offset: 1

Author

Alois P. Heinz, Jul 31 2013

Keywords

• nonn
• tabl

Examples

			:   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;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

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.

Programs

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 *)