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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.)