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Number of unlabeled rooted trees with n vertices whose non-leaf terminal subtrees are all different.
9

%I #4 Mar 22 2019 00:33:08

%S 1,1,2,4,8,17,37,83,189,436,1014,2373,5578,13156,31104,73665,174665,

%T 414427,983606,2334488

%N Number of unlabeled rooted trees with n vertices whose non-leaf terminal subtrees are all different.

%C The Matula-Goebel numbers of these trees are given by A324935.

%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(oo)) (o(ooo))

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

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

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

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

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

%e ((oo(o)))

%e (o((oo)))

%e (o(o(o)))

%e (oo((o)))

%e ((((oo))))

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

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

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

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

%t durt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[durt/@ptn]],UnsameQ@@Cases[#,{__},{0,Infinity}]&],{ptn,IntegerPartitions[n-1]}];

%t Table[Length[durt[n]],{n,10}]

%Y Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Mar 21 2019