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A141268 Number of phylogenetic rooted trees with n unlabeled objects. 50
1, 2, 4, 11, 30, 96, 308, 1052, 3648, 13003, 47006, 172605, 640662, 2402388, 9082538, 34590673, 132566826, 510904724, 1978728356, 7697565819, 30063818314, 117840547815, 463405921002, 1827768388175, 7228779397588, 28661434308095, 113903170011006, 453632267633931 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Unlabeled analog of A005804 = Phylogenetic trees with n labels.

From Gus Wiseman, Jul 31 2018: (Start)

a(n) is the number of series-reduced rooted trees whose leaves form an integer partition of n. For example, the following are the a(4) = 11 series-reduced rooted trees whose leaves form an integer partition of 4.

  4,

  (13),

  (22),

  (112), (1(12)), (2(11)),

  (1111), (11(11)), (1(1(11))), (1(111)), ((11)(11)).

(End)

REFERENCES

Klein, Moshe, and A. Yu Khrennikov. "Recursion over partitions." P-Adic Numbers, Ultrametric Analysis, and Applications 6.4 (2014): 303-309; http://www.hamataraemet.org/wp-content/uploads/2014/10/Recursion-over-partitions-4.10.2014.pdf. (See sp_n)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

a(n) ~ c * d^n / n^(3/2), where d = 4.210216501727104448901818751..., c = 0.21649387167268793159311306... . - Vaclav Kotesovec, Sep 04 2014

EXAMPLE

For n=4 we have A141268(4)=11 because

Set(Set(Z),Set(Z),Set(Z,Z)),

Set(Set(Z),Set(Set(Z),Set(Z,Z))),

Set(Z,Z,Z,Z),

Set(Set(Z,Z),Set(Z,Z)),

Set(Set(Set(Z),Set(Z)),Set(Z,Z)),

Set(Set(Z),Set(Z),Set(Set(Z),Set(Z))),

Set(Set(Z),Set(Z),Set(Z),Set(Z)),

Set(Set(Z),Set(Set(Z),Set(Z),Set(Z))),

Set(Set(Set(Z),Set(Z)),Set(Set(Z),Set(Z))),

Set(Set(Z),Set(Z,Z,Z)),

Set(Set(Z),Set(Set(Z),Set(Set(Z),Set(Z))))

MAPLE

with(combstruct): A141268 := [H, {H=Union(Set(Z, card>=1), Set(H, card>=2))}, unlabelled]; seq(count(A141268, size=j), j=1..20);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-i*j, i-1)*binomial(a(i)+j-1, j), j=0..n/i)))

    end:

a:= n-> `if`(n<2, n, 1+b(n, n-1)):

seq(a(n), n=1..30);  # Alois P. Heinz, Jun 18 2018

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

t[n_]:=t[n]=If[PrimeQ[n], {n}, Join@@Table[Union[Sort/@Tuples[t/@fac]], {fac, Select[facs[n], Length[#]>1&]}]];

Table[Sum[Length[t[Times@@Prime/@ptn]], {ptn, IntegerPartitions[n]}], {n, 7}] (* Gus Wiseman, Jul 31 2018 *)

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]=1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 26 2018

CROSSREFS

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316655.

Sequence in context: A007719 A148156 A148157 * A135527 A215460 A148158

Adjacent sequences:  A141265 A141266 A141267 * A141269 A141270 A141271

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jun 20 2008

EXTENSIONS

Offset corrected and more terms from Alois P. Heinz, Apr 21 2012

STATUS

approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)