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A300354
Number of enriched p-trees of weight n with distinct leaves.
7
1, 1, 1, 2, 2, 3, 8, 8, 13, 17, 54, 56, 98, 125, 195, 500, 606, 921, 1317, 1912, 2635, 6667, 7704, 12142, 16958, 24891, 33388, 47792, 106494, 126475, 195475, 268736, 393179, 523775, 750251, 979518, 2090669, 2457315, 3759380, 5066524, 7420874, 9726501, 13935546
OFFSET
0,4
COMMENTS
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.
FORMULA
a(n) = Sum_{i=1..A000009(n)} A299203(A246867(n,i)).
EXAMPLE
The a(6) = 8 enriched p-trees with distinct leaves: 6, (42), (51), ((31)2), ((32)1), (3(21)), ((21)3), (321).
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
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}]]];
Table[Total[ept/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, 1, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 03 2018
STATUS
approved