A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680

Offset

1,2

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced phylogenetic rooted trees on n distinct labels.

Links

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

Example

The a(1) = 1 through a(4) = 18 rooted trees:
  (1)  (12)      (123)        (1234)
       ((1)(2))  ((1)(23))    ((1)(234))
                 ((2)(13))    ((12)(34))
                 ((3)(12))    ((13)(24))
                 ((1)(2)(3))  ((14)(23))
                              ((2)(134))
                              ((3)(124))
                              ((4)(123))
                              ((1)(2)(34))
                              ((1)(3)(24))
                              ((1)(4)(23))
                              ((2)(3)(14))
                              ((2)(4)(13))
                              ((3)(4)(12))
                              ((1)(2)(3)(4))
                              (((1)(2))((3)(4)))
                              (((1)(3))((2)(4)))
                              (((1)(4))((2)(3)))

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]], {mtn, Select[sps[m], Length[#]>1&]}], m];
Table[Length[Select[gug[Range[n]], SameQ@@Length/@Position[#, _Integer]&]], {n, 9}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n, k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n, k, b(n, k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i, k]))} \\ Andrew Howroyd, Oct 26 2018

Crossrefs

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.

Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.

Sequence in context: A173227 A099556 A057864 * A032273 A143522 A217389

Adjacent sequences: A320151 A320152 A320153 * A320155 A320156 A320157

Keyword

nonn

Author

Gus Wiseman, Oct 06 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018

Status

approved