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%I #5 Mar 03 2018 22:52:00
%S 1,1,1,2,2,3,8,8,13,17,54,56,98,125,195,500,606,921,1317,1912,2635,
%T 6667,7704,12142,16958,24891,33388,47792,106494,126475,195475,268736,
%U 393179,523775,750251,979518,2090669,2457315,3759380,5066524,7420874,9726501,13935546
%N Number of enriched p-trees of weight n with distinct leaves.
%C An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.
%F a(n) = Sum_{i=1..A000009(n)} A299203(A246867(n,i)).
%e The a(6) = 8 enriched p-trees with distinct leaves: 6, (42), (51), ((31)2), ((32)1), (3(21)), ((21)3), (321).
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t ept[q_]:=ept[q]=If[Length[q]===1,1,Total[Times@@@Map[ept,Join@@Function[sptn,Join@@@Tuples[Permutations/@GatherBy[sptn,Total]]]/@Select[sps[q],Length[#]>1&],{2}]]];
%t Table[Total[ept/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,30}]
%Y Cf. A000009, A000041, A063834, A196545, A246867, A273873, A281145, A289501, A290261, A294018, A296150, A299201, A299202, A299203, A300352, A300353, A300355.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 03 2018