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A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

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