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

Author

Gus Wiseman, Sep 17 2018

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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}]

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

Extensions

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