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A319312
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Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.
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37
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1, 3, 7, 22, 67, 242, 885, 3456, 13761, 56342, 234269, 989335, 4225341, 18231145, 79321931, 347676128, 1533613723, 6803017863, 30328303589, 135808891308, 610582497919, 2755053631909, 12472134557093, 56630659451541, 257841726747551, 1176927093597201
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OFFSET
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1,2
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COMMENTS
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Also the number of orderless tree-factorizations of Heinz numbers of integer partitions of n.
Also the number of phylogenetic trees on a multiset of labels summing to n.
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LINKS
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EXAMPLE
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The a(3) = 7 trees:
(3) (21) (111)
((1)(2)) ((1)(11))
((1)(1)(1))
((1)((1)(1)))
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
phyfacs[n_]:=Prepend[Join@@Table[Union[Sort/@Tuples[phyfacs/@f]], {f, Select[facs[n], Length[#]>1&]}], n];
Table[Sum[Length[phyfacs[Times@@Prime/@m]], {m, IntegerPartitions[n]}], {n, 6}]
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[]); for(n=1, n, v=concat(v, numbpart(n) + EulerT(concat(v, [0]))[n])); v} \\ Andrew Howroyd, Sep 18 2018
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CROSSREFS
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Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316653, A316654, A316656.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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