A358591 - OEIS

%I #10 Jan 01 2023 14:46:36

%S 0,0,2,17,94,464,2162,9743,42962,186584,801316,3412034,14430740,

%T 60700548,254180426,1060361147,4409342954,18285098288,75645143516,

%U 312286595342,1286827096964,5293833371408,21745951533236,89208948855542,365523293690804,1496048600896784

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

%H Andrew Howroyd, <a href="/A358591/b358591.txt">Table of n, a(n) for n = 1..100</a>

%e The a(3) = 2 and a(4) = 17 trees:

%e   ((o)(oo))  (((o))(ooo))

%e   (o(o)(o))  (((o)(ooo)))

%e              (((oo))(oo))

%e              (((oo)(oo)))

%e              ((o)((ooo)))

%e              ((o)(o(oo)))

%e              ((o)(oo(o)))

%e              ((o(o)(oo)))

%e              ((oo)(o(o)))

%e              ((oo(o)(o)))

%e              (o((o))(oo))

%e              (o((o)(oo)))

%e              (o(o)((oo)))

%e              (o(o)(o(o)))

%e              (o(o(o)(o)))

%e              (oo((o)(o)))

%e              (oo(o)((o)))

%t art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];

%t Table[Length[Select[art[n],Count[#,_[__],{0,Infinity}]==Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,2,15,2}]

%o (PARI) \\ Needs R(n,f) defined in A358589.

%o 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

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

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

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

%Y These trees are ranked by A358592.

%Y A000081 counts rooted trees, ordered A000108.

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

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

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Nov 23 2022

%E Terms a(10) and beyond from _Andrew Howroyd_, Jan 01 2023