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%I #6 Mar 18 2019 08:15:10
%S 1,1,2,3,6,11,26,52,119,266,618,1432,3402,8093,19505,47228,115244,
%T 282529,696388,1723400
%N Number of recursively anti-transitive rooted trees with n nodes.
%C An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.
%e The a(1) = 1 through a(6) = 11 recursively anti-transitive rooted trees:
%e o (o) (oo) (ooo) (oooo) (ooooo)
%e ((o)) ((oo)) ((ooo)) ((oooo))
%e (((o))) (((oo))) (((ooo)))
%e ((o)(o)) ((o)(oo))
%e (o((o))) (o((oo)))
%e ((((o)))) (oo((o)))
%e ((((oo))))
%e (((o)(o)))
%e ((o((o))))
%e (o(((o))))
%e (((((o)))))
%t nallt[n_]:=Select[Union[Sort/@Join@@(Tuples[nallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@#,#]=={}&];
%t Table[Length[nallt[n]],{n,10}]
%Y Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
%Y Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
%Y Cf. A324838, A324840, A324844.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Mar 17 2019
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