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

Author

Gus Wiseman, Oct 07 2018

Keywords

• nonn

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.

Examples

			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))))
		

Links

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

Crossrefs

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

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

Programs

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}]

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

Extensions

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