The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #9 Jan 13 2024 16:13:28
%S 1,1,1,4,7,31,66,302,715,3313,8398,39095,104006,484706,1337220,
%T 6227730,17678835,82204045,238819350,1108202513,3282060210,
%U 15195242478,45741281820,211271435479,644952073662,2971835602526,9183676536076,42217430993002,131873975875180,604834233372884
%N Number of ordered rooted trees with n nodes, at least half of which are leaves.
%H Andrew Howroyd, <a href="/A358586/b358586.txt">Table of n, a(n) for n = 1..1000</a>
%F From _Andrew Howroyd_, Jan 13 2024: (Start)
%F a(n) = Sum_{k=1..floor(n/2)} A001263(n-1, k) for n >= 2.
%F a(2*n) = (A000108(2*n-1) + A000891(n-1))/2 for n >= 1;
%F a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)
%e The a(1) = 1 through a(5) = 7 ordered trees:
%e o (o) (oo) (ooo) (oooo)
%e ((o)o) ((o)oo)
%e ((oo)) ((oo)o)
%e (o(o)) ((ooo))
%e (o(o)o)
%e (o(oo))
%e (oo(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}]>=Count[#,_[__],{0,Infinity}]&]],{n,1,10}]
%o (PARI) a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) + if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ _Andrew Howroyd_, Jan 13 2024
%Y For equality we have A000891, unordered A185650.
%Y Odd-indexed terms appear to be A065097.
%Y The unordered version is A358583.
%Y The opposite is the same, unordered A358584.
%Y The strict case is A358585, unordered A358581.
%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 A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
%Y Cf. A109129, A342507, A358371, A358579, A358580, A358582, A358584, A358588.
%K nonn
%O 1,4
%A _Gus Wiseman_, Nov 24 2022
%E a(16) onwards from _Andrew Howroyd_, Jan 13 2024