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A320178 Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 4 11:32 EDT 2020. Contains 334825 sequences. (Running on oeis4.)