A320178 Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

1, 2, 4, 8, 19, 53, 151, 459, 1445, 4634, 15154, 50253, 168607, 571212, 1951588, 6715575, 23255444, 80978697, 283373024, 995995996, 3514614634, 12446666967, 44222390525, 157587392768, 563096832839, 2017121728223, 7242436444030, 26059512879605, 93952946906117

Offset

1,2

Comments

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

In an identity tree, all branches directly under any given node are different.

Links

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

Example

The a(1) = 1 through a(5) = 19 rooted trees:
  (1)  (2)   (3)        (4)             (5)
       (11)  (111)      (22)            (11111)
             ((1)(2))   (1111)          ((1)(4))
             ((1)(11))  ((1)(3))        ((2)(3))
                        ((2)(11))       ((1)(22))
                        ((1)(111))      ((3)(11))
                        ((1)((1)(2)))   ((2)(111))
                        ((1)((1)(11)))  ((1)(1111))
                                        ((11)(111))
                                        ((1)(2)(11))
                                        ((1)((1)(3)))
                                        ((2)((1)(2)))
                                        ((11)((1)(2)))
                                        ((1)((2)(11)))
                                        ((2)((1)(11)))
                                        ((1)((1)(111)))
                                        ((11)((1)(11)))
                                        ((1)((1)((1)(2))))
                                        ((1)((1)((1)(11))))

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

Prog

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

Crossrefs

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320177.

Sequence in context: A099598 A269023 A173310 * A128816 A006897 A287025

Adjacent sequences: A320175 A320176 A320177 * A320179 A320180 A320181

Keyword

nonn

Author

Gus Wiseman, Oct 07 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

Status

approved