%I #9 Oct 26 2018 00:52:09
%S 1,2,3,6,9,19,31,63,110,215,391,773,1451,2879,5594,11173,22041,44136,
%T 87631,175155,348186,694013,1378911,2743955,5452833,10853541,21610732,
%U 43122952,86192274,172753293,347114772,699602332,1414033078,2866580670,5826842877,11874508385
%N Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.
%C 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.
%C Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.
%H Andrew Howroyd, <a href="/A320160/b320160.txt">Table of n, a(n) for n = 1..500</a>
%e The a(1) = 1 through a(6) = 19 rooted trees:
%e 1 2 3 4 5 6
%e (11) (12) (13) (14) (15)
%e (111) (22) (23) (24)
%e (112) (113) (33)
%e (1111) (122) (114)
%e ((11)(11)) (1112) (123)
%e (11111) (222)
%e ((11)(12)) (1113)
%e ((11)(111)) (1122)
%e (11112)
%e (111111)
%e ((11)(13))
%e ((11)(22))
%e ((12)(12))
%e ((11)(112))
%e ((12)(111))
%e ((11)(1111))
%e ((111)(111))
%e ((11)(11)(11))
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[mps[Sort[labs]],Length[#1]>1&]}]];
%t Table[Sum[Length[Select[phy2[ptn],SameQ@@Length/@Position[#,_Integer]&]],{ptn,IntegerPartitions[n]}],{n,8}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o 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
%Y Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.
%Y Cf. A316624, A320154, A320155, A320169, A320173, A320179.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 06 2018
%E Terms a(14) and beyond from _Andrew Howroyd_, Oct 25 2018