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%I #9 Aug 08 2018 07:53:16
%S 1,1,1,1,2,2,4,4,7,8,12,14,21,24,34,42,55,67,91,109,144,177,228,281,
%T 366,448,579,720,916,1142
%N Number of locally connected rooted trees with n nodes.
%C An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.
%H Gus Wiseman, <a href="/A317785/a317785.png">All 42 locally connected rooted trees with 16 nodes.</a>
%e The a(11) = 12 locally connected rooted trees:
%e ((((((((((o))))))))))
%e ((((((((o)(o))))))))
%e (((((((o))((o)))))))
%e ((((((o)))(((o))))))
%e (((((o))))((((o)))))
%e ((((((o)(o)(o))))))
%e (((((o))((o)(o)))))
%e ((((o))((o))((o))))
%e ((((o)(o)(o)(o))))
%e (((o))((o)(o)(o)))
%e (((o)(o))((o)(o)))
%e ((o)(o)(o)(o)(o))
%t multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
%t rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Length[csm[#]]==1]&],{ptn,IntegerPartitions[n-1]}]];
%t Table[Length[rurt[n]],{n,10}]
%Y Cf. A000081, A276625, A286518, A286520, A301700, A304714, A316473, A316475, A317077, A317078, A317708, A317787.
%K nonn,more
%O 1,5
%A _Gus Wiseman_, Aug 06 2018
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