%I #7 Sep 14 2018 15:58:51
%S 1,1,2,3,6,13,28,64,153,379,939,2385,6121,15871,41529,109509,290607,
%T 775842,2081874,5612176,15191329,41274052
%N Number of series-reduced locally nonintersecting rooted identity trees whose leaves form an integer partition of n.
%C A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty. It is an identity tree if no branch appears multiple times under the same root.
%e The a(6) = 13 trees:
%e (1(1(1(12))))
%e (1(1(13)))
%e (1(2(12)))
%e (2(1(12)))
%e (12(12))
%e (1(14))
%e (1(23))
%e (2(13))
%e (3(12))
%e (123)
%e (15)
%e (24)
%e 6
%e Examples of series-reduced rooted identity trees that are not locally nonintersecting are ((12)(13)) and ((12)(1(12))).
%t nonintQ[u_]:=Intersection@@u=={};
%t nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,nonintQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
%t Table[Length[nms[n]],{n,15}]
%Y Cf. A000081, A000669, A001678, A141268, A292504, A316500, A316651, A316652, A316655, A316696, A316768.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Jul 12 2018
%E a(21)-a(22) from _Robert Price_, Sep 14 2018