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Number of fully anti-transitive rooted trees with n nodes.
14

%I #8 Jun 20 2020 02:46:44

%S 1,1,2,3,6,11,27,60,152,376,968,2492,6549,17259,46000,123214,332304,

%T 900406,2451999,6703925

%N Number of fully anti-transitive rooted trees with n nodes.

%C An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

%e The a(1) = 1 through a(6) = 11 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 rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];

%t Table[Length[Select[rtall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]

%Y Cf. A000081, A279861, A290689, A304360, A306844, A318185.

%Y Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324769, A324770.

%Y Cf. A324838, A324840, A324844, A324846.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Mar 17 2019

%E a(17)-a(20) from _Jinyuan Wang_, Jun 20 2020