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Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.
20

%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