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A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

%I A331783 #6 Feb 02 2020 09:03:41
%S A331783 1,1,2,4,8,17,37,83,191,450,1076,2610,6404,15875,39676,99880,253016,
%T A331783 644524,1649918,4242226
%N A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.
%C A331783 Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.
%e A331783 The a(1) = 1 through a(6) = 17 trees:
%e A331783   o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
%e A331783           ((o))  ((oo))   ((ooo))    ((oooo))
%e A331783                  (o(o))   (o(oo))    (o(ooo))
%e A331783                  (((o)))  (oo(o))    (oo(oo))
%e A331783                           (((oo)))   (ooo(o))
%e A331783                           ((o(o)))   (((ooo)))
%e A331783                           (o((o)))   ((o(oo)))
%e A331783                           ((((o))))  ((oo(o)))
%e A331783                                      (o((oo)))
%e A331783                                      (o(o(o)))
%e A331783                                      (oo((o)))
%e A331783                                      ((((oo))))
%e A331783                                      (((o(o))))
%e A331783                                      ((o)((o)))
%e A331783                                      ((o((o))))
%e A331783                                      (o(((o))))
%e A331783                                      (((((o)))))
%t A331783 disjunsQ[u_]:=Length[u]==1||UnsameQ@@DeleteCases[u,{}]&&Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A331783 ldrsi[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[ldrsi/@c]]]/@IntegerPartitions[n-1],disjunsQ]];
%t A331783 Table[Length[ldrsi[n]],{n,10}]
%Y A331783 The lone-child-avoiding case is A212804.
%Y A331783 The identity tree version is A316471.
%Y A331783 The Matula-Goebel numbers of these trees are given by A331682.
%Y A331783 Identity trees are A004111.
%Y A331783 Semi-identity trees are A306200.
%Y A331783 Locally disjoint rooted trees are A316473.
%Y A331783 Matula-Goebel numbers of locally disjoint semi-identity trees are A316494.
%Y A331783 Cf. A000081, A306202, A316475, A316495, A316694, A331683, A331684, A331686.
%K A331783 nonn,more
%O A331783 1,3
%A A331783 _Gus Wiseman_, Jan 31 2020