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A316655
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Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.
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16
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0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712
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OFFSET
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1,5
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COMMENTS
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A rooted tree is series-reduced if every non-leaf node has at least two branches.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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LINKS
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FORMULA
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EXAMPLE
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Sequence of sets of trees begins:
1:
2: 1
3: (11)
4: (12)
5: (1(11)), (111)
6: (1(12)), (2(11)), (112)
7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)
8: (1(23)), (2(13)), (3(12)), (123)
9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1, m, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]];
Table[Length[gro[Flatten[MapIndexed[Table[#2, {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]]], {n, 20}]
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CROSSREFS
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Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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