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A358591
Number of 2n-node rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.
10
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
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.
Sequence in context: A372189 A219757 A297727 * A002645 A100268 A163790
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 23 2022
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved