A320174 Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

1, 3, 6, 19, 55, 200, 713, 2740, 10651, 42637, 173012, 713280, 2972389, 12514188, 53119400, 227140464, 977382586, 4229274235, 18391269922, 80330516578, 352269725526, 1550357247476, 6845517553493, 30316222112019, 134626183784975, 599341552234773, 2674393679352974

Offset

1,2

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches.

Links

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

Example

The a(1) = 1 through a(4) = 19 trees:
  (1)  (2)       (3)            (4)
       (11)      (111)          (22)
       ((1)(1))  ((1)(2))       (1111)
                 ((1)(11))      ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((2)(11))
                                ((1)(111))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)(1)(11))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)((1)(11)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

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]]]];
dot[m_]:=If[SameQ@@m, Prepend[#, m], #]&[Join@@Table[Union[Sort/@Tuples[dot/@p]], {p, Select[mps[m], Length[#]>1&]}]];
Table[Length[Join@@Table[dot[m], {m, IntegerPartitions[n]}]], {n, 10}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Crossrefs

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320175.

Sequence in context: A148571 A345400 A148572 * A248603 A332344 A345244

Adjacent sequences: A320171 A320172 A320173 * A320175 A320176 A320177

Keyword

nonn

Author

Gus Wiseman, Oct 07 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved