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A319312
Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.
37
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2018
EXTENSIONS
Terms a(14) and beyond from Andrew Howroyd, Sep 18 2018
STATUS
approved