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A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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).
LINKS
FORMULA
a(prime(n)) = A000669(n).
a(2^n) = A000311(n).
EXAMPLE
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)
MATHEMATICA
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}]
CROSSREFS
Sequence in context: A258861 A171038 A023395 * A318848 A193798 A101409
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 09 2018
EXTENSIONS
a(37)-a(40) from Robert Price, Sep 13 2018
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)