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%I #9 Dec 30 2022 21:38:34
%S 0,0,1,1,5,7,28,48,176,336,1179,2420,8269,17855,59832,134289,443407,
%T 1025685,3346702,7933161,25632265,62000170,198670299,488801159,
%U 1555187172,3882403641,12276230777,31034921462,97601239282,249471619165,780790439063,2015194486878
%N Number of rooted trees with n nodes, most of which are not leaves.
%H Andrew Howroyd, <a href="/A358582/b358582.txt">Table of n, a(n) for n = 1..200</a>
%F A358581(n) + A358584(n) = A000081(n).
%F A358582(n) + A358583(n) = A000081(n).
%F a(n) = Sum_{k=1..floor((n-1)/2)} A055277(n, k). - _Andrew Howroyd_, Dec 30 2022
%e The a(3) = 1 through a(6) = 7 trees:
%e ((o)) (((o))) (((oo))) ((((oo))))
%e ((o)(o)) (((o)(o)))
%e ((o(o))) (((o(o))))
%e (o((o))) ((o)((o)))
%e ((((o)))) ((o((o))))
%e (o(((o))))
%e (((((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}]&]],{n,0,10}]
%o (PARI) \\ See A358584 for R(n).
%o seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..(n-1)\2]))} \\ _Andrew Howroyd_, Dec 30 2022
%Y For equality we have A185650 aerated, ranked by A358578.
%Y The opposite version is A358581, non-strict A358583.
%Y The non-strict version is A358584.
%Y The ordered version is A358585, odd-indexed terms A065097.
%Y A000081 counts rooted trees, ordered A000108.
%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
%Y A358589 counts square trees, ranked by A358577, ordered A358590.
%Y Cf. A000891, A034781, A109129, A342507, A358579, A358580, A358586, A358591.
%K nonn
%O 1,5
%A _Gus Wiseman_, Nov 23 2022
%E Terms a(19) and beyond from _Andrew Howroyd_, Dec 30 2022
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