OFFSET
1,2
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.
Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
EXAMPLE
The a(1) = 1 through a(6) = 19 rooted trees:
1 2 3 4 5 6
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(1111) (122) (114)
((11)(11)) (1112) (123)
(11111) (222)
((11)(12)) (1113)
((11)(111)) (1122)
(11112)
(111111)
((11)(13))
((11)(22))
((12)(12))
((11)(112))
((12)(111))
((11)(1111))
((111)(111))
((11)(11)(11))
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
phy2[labs_]:=If[Length[labs]==1, labs, Union@@Table[Sort/@Tuples[phy2/@ptn], {ptn, Select[mps[Sort[labs]], Length[#1]>1&]}]];
Table[Sum[Length[Select[phy2[ptn], SameQ@@Length/@Position[#, _Integer]&]], {ptn, IntegerPartitions[n]}], {n, 8}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, 1), v=vector(n)); while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 06 2018
EXTENSIONS
Terms a(14) and beyond from Andrew Howroyd, Oct 25 2018
STATUS
approved