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%I #11 Aug 05 2018 20:42:35
%S 1,1,2,4,7,13,27,55,125,278,650,1510,3624,8655,21017,51212,125857,
%T 310581,770767,1920226
%N Number of uniform relatively prime rooted trees with n nodes.
%C An unlabeled rooted tree is uniform and relatively prime iff either it is a single node or a single node with a single uniform relatively prime branch, or the branches of the root have empty intersection (relatively prime) and equal multiplicities (uniform) and are themselves uniform relatively prime trees.
%H A. David Christopher and M. Davamani Christober, <a href="http://emis.impa.br/EMIS/journals/GMN/yahoo_site_admin/assets/docs/1_GMN-2492-V13N2.77213831.pdf">Relatively Prime Uniform Partitions</a>, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.
%e The a(6) = 13 uniform relatively prime rooted trees:
%e (((((o)))))
%e ((((oo))))
%e (((o(o))))
%e (((ooo)))
%e ((o((o))))
%e ((o(oo)))
%e ((oooo))
%e (o(((o))))
%e (o((oo)))
%e (o(o(o)))
%e (o(ooo))
%e ((o)((o)))
%e (ooooo)
%t purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,And[SameQ@@Length/@Split[#],Intersection@@#=={}]]&],{ptn,IntegerPartitions[n-1]}]];
%t Table[Length[purt[n]],{n,20}]
%Y Cf. A000081, A001190, A004111, A072774, A301700, A317588.
%Y Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317717.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Aug 05 2018
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