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%I #5 Nov 18 2022 23:37:01
%S 1,1,2,3,7,17,47,117,321,895,2556,7331,21435,63116,187530
%N Number of recursively bi-anti-transitive ordered rooted trees with n nodes.
%C We define an unlabeled ordered rooted tree to be recursively bi-anti-transitive if there are no two branches of the same node such that one is a branch of the other.
%e The a(1) = 1 through a(6) = 17 trees:
%e o (o) (oo) (ooo) (oooo) (ooooo)
%e ((o)) ((oo)) ((ooo)) ((oooo))
%e (((o))) (((o))o) (((o))oo)
%e (((oo))) (((oo))o)
%e ((o)(o)) (((ooo)))
%e (o((o))) ((o)(oo))
%e ((((o)))) ((oo)(o))
%e (o((o))o)
%e (o((oo)))
%e (oo((o)))
%e ((((o)))o)
%e ((((o))o))
%e ((((oo))))
%e (((o)(o)))
%e ((o((o))))
%e (o(((o))))
%e (((((o)))))
%t aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[Select[aot[n],FreeQ[#,{___,x_,___,{___,x_,___},___}|{___,{___,x_,___},___,x_,___}]&]],{n,10}]
%Y The unordered version is A324765, ranked by A324766.
%Y The directed version is A358455.
%Y A000108 counts ordered rooted trees, unordered A000081.
%Y A306844 counts anti-transitive rooted trees.
%Y A358453 counts transitive ordered trees, unordered A290689.
%Y Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, A324844.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Nov 18 2022
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