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A290689 Number of transitive rooted trees with n nodes.

%I A290689 #32 Nov 18 2024 08:55:25
%S A290689 1,1,1,2,3,5,8,13,21,34,55,88,143,229,370,592,955,1527,2457,3929,6304,
%T A290689 10081
%N A290689 Number of transitive rooted trees with n nodes.
%C A290689 A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.
%C A290689 Regarding the notation, a rooted tree is a finite multiset of rooted trees. For example, the rooted tree (o(o)(oo)) is short for {{},{{}},{{},{}}}. Each "o" is a leaf. Each pair of parentheses corresponds to a non-leaf node (such as the root). Its contents "(...)" represent a branch. - _Gus Wiseman_, Nov 16 2024
%H A290689 Robert P. P. McKone, <a href="/A290689/a290689.txt">The transitive rooted trees with n=1 to n=22 nodes</a>.
%e A290689 The a(7) = 8 7-node transitive rooted trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).
%t A290689 nn=18;
%t A290689 rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
%t A290689 Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]
%Y A290689 The restriction to identity trees (A004111) is A279861, ranks A290760.
%Y A290689 These trees are ranked by A290822.
%Y A290689 The anti-transitive version is A306844, ranks A324758.
%Y A290689 The totally transitive case is A318185 (by leaves A318187), ranks A318186.
%Y A290689 A version for integer partitions is A324753, for subsets A324736.
%Y A290689 The ordered version is A358453, ranks A358457, undirected A358454.
%Y A290689 Cf. A000081, A001192, A001678, A324770, A324969, A325705, A358455, A358460.
%K A290689 nonn,more
%O A290689 1,4
%A A290689 _Gus Wiseman_, Oct 19 2017
%E A290689 a(20) from _Robert Price_, Sep 13 2018
%E A290689 a(21)-a(22) from _Robert P. P. McKone_, Dec 16 2023