A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1

Offset: 1

Author

Alois P. Heinz, Jul 08 2014

Keywords

• nonn
• tabl

Examples

			The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
  :    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  :
  :         :             :       :     :
  : ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,  1;
  0, 1,  2,  1,  1;
  0, 1,  2,  2,  1,  1;
  0, 1,  4,  3,  2,  1, 1;
  0, 1,  4,  5,  3,  2, 1, 1;
  0, 1,  7,  7,  6,  3, 2, 1, 1;
  0, 1,  8, 12,  8,  6, 3, 2, 1, 1;
  0, 1, 12, 18, 15,  9, 6, 3, 2, 1, 1;
  0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007(n-1), A000012 (for n>0), A002865(n-1) (for n>2), A048808, A048809, A048810, A048811, A048812, A048813, A048814, A048815.

T(2n+1,n) gives A074045.

Row sums give A048816.

Programs

Maple

with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
      add(add(`if`(d
				

Mathematica

T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[dJean-François Alcover, Jan 28 2015, after Alois P. Heinz *)