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%I #4 Jul 12 2018 20:25:00
%S 0,1,1,0,1,1,1,0,0,1,1,2,1,1,1,0,1,2,1,2,1,1,1,4,0,1,0,2,1,4,1,0,1,1,
%T 1,6,1,1,1,4,1,4,1,2,2,1,1,8,0,2,1,2,1,4,1,4,1,1,1,12,1,1,2,0,1,4,1,2,
%U 1,4,1,17,1,1,2,2,1,4,1,8,0,1,1,12,1,1
%N Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number 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.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e The a(36) = 6 trees:
%e (1(2(12)))
%e (2(1(12)))
%e (1(122))
%e (2(112))
%e (12(12))
%e (1122)
%t sps[{}]:={{}};
%t sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>Intersection@@q=={},{0,Infinity}]&]],{n,100}]
%Y Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316503, A316651, A316652, A316655.
%K nonn
%O 1,12
%A _Gus Wiseman_, Jul 12 2018
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