login
Number of rooted trees with n vertices with no proper terminal subtree appearing at only one position.
0

%I #6 May 17 2019 22:06:25

%S 1,0,1,1,2,2,5,5,11,13,27,30,69,76,168

%N Number of rooted trees with n vertices with no proper terminal subtree appearing at only one position.

%C The Matula-Goebel numbers of these trees are given by A325661.

%e The a(4) = 1 through a(9) = 11 rooted trees:

%e (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)

%e ((o)(o)) (o(o)(o)) ((oo)(oo)) (o(oo)(oo)) ((ooo)(ooo))

%e (oo(o)(o)) (ooo(o)(o)) (oo(oo)(oo))

%e ((o)(o)(o)) (o(o)(o)(o)) (oooo(o)(o))

%e (((o))((o))) (o((o))((o))) (oo(o)(o)(o))

%e (((oo))((oo)))

%e ((o)(o)(o)(o))

%e ((o(o))(o(o)))

%e (oo((o))((o)))

%e ((o)((o))((o)))

%e ((((o)))(((o))))

%t urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];

%t Table[Length[Select[urt[n],!MemberQ[Length/@Split[Sort[Extract[#,Most[Position[#,_List]]]]],1]&]],{n,15}]

%Y Cf. A000081, A001694, A004111, A290689, A306844, A317713, A324936, A324971, A325661.

%K nonn,more

%O 1,5

%A _Gus Wiseman_, May 17 2019