A358591 Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

0, 0, 2, 17, 94, 464, 2162, 9743, 42962, 186584, 801316, 3412034, 14430740, 60700548, 254180426, 1060361147, 4409342954, 18285098288, 75645143516, 312286595342, 1286827096964, 5293833371408, 21745951533236, 89208948855542, 365523293690804, 1496048600896784

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Example

The a(3) = 2 and a(4) = 17 trees:
  ((o)(oo))  (((o))(ooo))
  (o(o)(o))  (((o)(ooo)))
             (((oo))(oo))
             (((oo)(oo)))
             ((o)((ooo)))
             ((o)(o(oo)))
             ((o)(oo(o)))
             ((o(o)(oo)))
             ((oo)(o(o)))
             ((oo(o)(o)))
             (o((o))(oo))
             (o((o)(oo)))
             (o(o)((oo)))
             (o(o)(o(o)))
             (o(o(o)(o)))
             (oo((o)(o)))
             (oo(o)((o)))

Mathematica

art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n], Count[#, _[__], {0, Infinity}]==Count[#, {}, {0, Infinity}]==Depth[#]-1&]], {n, 2, 15, 2}]

Prog

(PARI) \\ Needs R(n, f) defined in A358589.
seq(n) = {Vecrev(R(2*n, (h, p)->if(h<=n, x^h*polcoef(polcoef(p, 2*h, x), h, y))), -n)} \\ Andrew Howroyd, Jan 01 2023

Crossrefs

For leaves = internals we have A185650 aerated, ranked by A358578.

For height = internals we have A358587, ranked by A358576, ordered A358588.

For height = leaves we have A358589, ranked by A358577, ordered A358590.

These trees are ranked by A358592.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height, ordered A080936.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

A358575 counts rooted trees by nodes and internal nodes, ordered A090181.

Cf. A065097, A109082, A109129, A342507, A358552, A358581-A358586.

Sequence in context: A081744 A219757 A297727 * A002645 A100268 A163790

Adjacent sequences: A358588 A358589 A358590 * A358592 A358593 A358594

Keyword

nonn

Author

Gus Wiseman, Nov 23 2022

Extensions

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

Status

approved