cp's OEIS Frontend

A317785 Number of locally connected rooted trees with n nodes.

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