cp's OEIS Frontend

A358588 Number of n-node ordered rooted trees of height equal to the number of internal (non-leaf) nodes.

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