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)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Moshe Klein and A. Yu Khrennikov, Recursion over partitions</a, P-Adic Numbers, Ultrametric Analysis, and Applications 6.4 (2014): 303-309. See sp_n.
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 *)
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 *)
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
KEYWORD
nonn
AUTHOR
Thomas Wieder, Jun 20 2008
EXTENSIONS
Offset corrected and more terms from Alois P. Heinz, Apr 21 2012
STATUS
approved