A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 1, 0, 1, 5, 6, 5, 3, 2, 1, 1, 1, 0, 1, 5, 9, 7, 5, 3, 2, 1, 1, 1, 0, 1, 7, 12, 12, 8, 5, 3, 2, 1, 1, 1, 0, 1, 8, 17, 17, 13, 8, 5, 3, 2, 1, 1, 1

Offset

1,18

Comments

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root.

Links

Alois P. Heinz, Rows n = 1..200, flattened

Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.

Example

Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  2  1  1  1
  0  1  2  2  1  1  1
  0  1  3  3  2  1  1  1
  0  1  3  4  3  2  1  1  1
  0  1  5  6  5  3  2  1  1  1
  0  1  5  9  7  5  3  2  1  1  1
  0  1  7 12 12  8  5  3  2  1  1  1
  0  1  8 17 17 13  8  5  3  2  1  1  1
  0  1 10 25 26 20 14  8  5  3  2  1  1  1
  0  1 12 34 39 31 21 14  8  5  3  2  1  1  1
The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
  ((ooooooo))   (((oooooo)))    ((((ooooo))))    (((((oooo)))))
  ((o)(ooooo))  (((o)(oooo)))   ((((o)(ooo))))   (((((o)(oo)))))
  ((oo)(oooo))  (((oo)(ooo)))   ((((o))((oo))))
                (((o))((ooo)))

Mathematica

ubk[n_, k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#, k-1]&/@ptn]], UnsameQ@@DeleteCases[#, {}]&], {ptn, IntegerPartitions[n-1]}], SameQ[k, ##]&@@Length/@Position[#, {}]&];
Table[Length[ubk[n, k]], {n, 1, 10}, {k, 0, n-1}]

Crossrefs

Row sums give A306201.

T(2n-1,n) gives A306274.

Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.

Sequence in context: A278287 A227752 A316771 * A035212 A318133 A068029

Adjacent sequences: A306266 A306267 A306268 * A306270 A306271 A306272

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 01 2019

Status

approved