A319312 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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

Offset

1,2

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Example

The a(3) = 7 trees:
  (3)    (21)        (111)
       ((1)(2))    ((1)(11))
                  ((1)(1)(1))
                 ((1)((1)(1)))

Mathematica

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}]

Prog

(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

Crossrefs

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316653, A316654, A316656.

Sequence in context: A242566 A148686 A148687 * A325213 A148688 A045610

Adjacent sequences: A319309 A319310 A319311 * A319313 A319314 A319315

Keyword

nonn

Author

Gus Wiseman, Sep 17 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 18 2018

Status

approved