A141268 Number of phylogenetic rooted trees with n unlabeled objects.

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

Offset: 1

Author

Thomas Wieder, Jun 20 2008

Keywords

• nonn

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)

Examples

			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))))
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Moshe Klein and A. Yu. Khrennikov, Recursion over partitions, P-Adic Numbers, Ultrametric Analysis, and Applications 6.4 (2014): 303-309. See sp_n.

Crossrefs

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

Programs

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 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n-i*j, i-1]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, 1 + b[n, n-1]];
Array[a, 30] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

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

Formula

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

Extensions

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