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A141268 Number of phylogenetic rooted trees with n unlabeled objects.

%I A141268 #33 Jan 08 2025 10:27:34
%S A141268 1,2,4,11,30,96,308,1052,3648,13003,47006,172605,640662,2402388,
%T A141268 9082538,34590673,132566826,510904724,1978728356,7697565819,
%U A141268 30063818314,117840547815,463405921002,1827768388175,7228779397588,28661434308095,113903170011006,453632267633931
%N A141268 Number of phylogenetic rooted trees with n unlabeled objects.
%C A141268 Unlabeled analog of A005804 = Phylogenetic trees with n labels.
%C A141268 From _Gus Wiseman_, Jul 31 2018: (Start)
%C A141268 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.
%C A141268   4,
%C A141268   (13),
%C A141268   (22),
%C A141268   (112), (1(12)), (2(11)),
%C A141268   (1111), (11(11)), (1(1(11))), (1(111)), ((11)(11)).
%C A141268 (End)
%H A141268 Alois P. Heinz, <a href="/A141268/b141268.txt">Table of n, a(n) for n = 1..1000</a>
%H A141268 Moshe Klein and A. Yu. Khrennikov, <a href="https://doi.org/10.1134/S2070046614040050">Recursion over partitions</a>, P-Adic Numbers, Ultrametric Analysis, and Applications 6.4 (2014): 303-309. See sp_n.
%F A141268 a(n) ~ c * d^n / n^(3/2), where d = 4.210216501727104448901818751..., c = 0.21649387167268793159311306... . - _Vaclav Kotesovec_, Sep 04 2014
%e A141268 For n=4 we have A141268(4)=11 because
%e A141268 Set(Set(Z),Set(Z),Set(Z,Z)),
%e A141268 Set(Set(Z),Set(Set(Z),Set(Z,Z))),
%e A141268 Set(Z,Z,Z,Z),
%e A141268 Set(Set(Z,Z),Set(Z,Z)),
%e A141268 Set(Set(Set(Z),Set(Z)),Set(Z,Z)),
%e A141268 Set(Set(Z),Set(Z),Set(Set(Z),Set(Z))),
%e A141268 Set(Set(Z),Set(Z),Set(Z),Set(Z)),
%e A141268 Set(Set(Z),Set(Set(Z),Set(Z),Set(Z))),
%e A141268 Set(Set(Set(Z),Set(Z)),Set(Set(Z),Set(Z))),
%e A141268 Set(Set(Z),Set(Z,Z,Z)),
%e A141268 Set(Set(Z),Set(Set(Z),Set(Set(Z),Set(Z))))
%p A141268 with(combstruct): A141268 := [H, {H=Union(Set(Z,card>=1),Set(H,card>=2))}, unlabelled]; seq(count(A141268, size=j), j=1..20);
%p A141268 # second Maple program:
%p A141268 b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A141268       add(b(n-i*j, i-1)*binomial(a(i)+j-1, j), j=0..n/i)))
%p A141268     end:
%p A141268 a:= n-> `if`(n<2, n, 1+b(n, n-1)):
%p A141268 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jun 18 2018
%t A141268 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A141268 t[n_]:=t[n]=If[PrimeQ[n],{n},Join@@Table[Union[Sort/@Tuples[t/@fac]],{fac,Select[facs[n],Length[#]>1&]}]];
%t A141268 Table[Sum[Length[t[Times@@Prime/@ptn]],{ptn,IntegerPartitions[n]}],{n,7}] (* _Gus Wiseman_, Jul 31 2018 *)
%t A141268 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A141268      Sum[b[n-i*j, i-1]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
%t A141268 a[n_] := If[n < 2, n, 1 + b[n, n-1]];
%t A141268 Array[a, 30] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%o A141268 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A141268 seq(n)={my(v=vector(n)); for(n=1, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 26 2018
%Y A141268 Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316655.
%K A141268 nonn
%O A141268 1,2
%A A141268 _Thomas Wieder_, Jun 20 2008
%E A141268 Offset corrected and more terms from _Alois P. Heinz_, Apr 21 2012