%I #12 Apr 15 2024 04:57:31
%S 0,0,0,0,1,8,41,171,633,2171,7070,22195,67830,203130,598806,1743258,
%T 5023711,14356226,40737383,114904941,322432215,900707165,2506181060,
%U 6948996085,19207795836,52944197508,145567226556,399314965956,1093107693133,2986640695436
%N Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.
%H Andrew Howroyd, <a href="/A358588/b358588.txt">Table of n, a(n) for n = 1..200</a>
%F Conjectures from _Chai Wah Wu_, Apr 14 2024: (Start)
%F a(n) = 9*a(n-1) - 32*a(n-2) + 58*a(n-3) - 58*a(n-4) + 32*a(n-5) - 9*a(n-6) + a(n-7) for n > 7.
%F G.f.: x^5*(-x^2 + x - 1)/((x - 1)^3*(x^2 - 3*x + 1)^2). (End)
%e The a(5) = 1 and a(6) = 8 ordered trees:
%e ((o)(o)) ((o)(o)o)
%e ((o)(oo))
%e ((o)o(o))
%e ((oo)(o))
%e (o(o)(o))
%e (((o))(o))
%e (((o)(o)))
%e ((o)((o)))
%t aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[Select[aot[n],Count[#,_[__],{0,Infinity}]==Depth[#]-1&]],{n,1,10}]
%o (PARI) \\ Needs R(n,f) defined in A358590.
%o seq(n) = {Vec(R(n, (h,p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ _Andrew Howroyd_, Jan 01 2023
%Y For leaves instead of height we have A000891, unordered A185650 aerated.
%Y The unordered version is A358587, ranked by A358576.
%Y For leaves instead of internal nodes we have A358590, unordered A358589.
%Y A000108 counts ordered rooted trees, unordered A000081.
%Y A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
%Y A080936 counts ordered rooted trees by nodes and height, unordered A034781.
%Y A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
%Y Cf. A065097, A342507, A358552, A358371, A358578, A358579, A358586, A358591.
%K nonn
%O 1,6
%A _Gus Wiseman_, Nov 25 2022
%E Terms a(16) and beyond from _Andrew Howroyd_, Jan 01 2023