%I #5 Feb 01 2020 14:39:52
%S 1,2,4,12,29,93,249,803,2337,7480,23130,77372,247598,834507,2762222
%N Number of locally disjoint enriched p-trees of weight n.
%C A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.
%e The a(1) = 1 through a(4) = 12 enriched p-trees:
%e 1 2 3 4
%e (11) (21) (22)
%e (111) (31)
%e ((11)1) (211)
%e (1111)
%e ((11)2)
%e ((21)1)
%e (2(11))
%e ((11)11)
%e ((111)1)
%e (((11)1)1)
%e ((11)(11))
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
%t Table[Length[ldep[n]],{n,10}]
%Y The orderless version is A316696.
%Y The identity case is A331684.
%Y P-trees are A196545.
%Y Enriched p-trees are A289501.
%Y Locally disjoint identity trees are A316471.
%Y Enriched identity p-trees are A331875.
%Y Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Jan 31 2020