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 A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}. 12
 1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680 (list; graph; refs; listen; history; text; internal format)
 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 phylogenetic rooted trees on n distinct labels. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 EXAMPLE The a(1) = 1 through a(4) = 18 rooted trees:   (1)  (12)      (123)        (1234)        ((1)(2))  ((1)(23))    ((1)(234))                  ((2)(13))    ((12)(34))                  ((3)(12))    ((13)(24))                  ((1)(2)(3))  ((14)(23))                               ((2)(134))                               ((3)(124))                               ((4)(123))                               ((1)(2)(34))                               ((1)(3)(24))                               ((1)(4)(23))                               ((2)(3)(14))                               ((2)(4)(13))                               ((3)(4)(12))                               ((1)(2)(3)(4))                               (((1)(2))((3)(4)))                               (((1)(3))((2)(4)))                               (((1)(4))((2)(3))) MATHEMATICA sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]], {mtn, Select[sps[m], Length[#]>1&]}], m]; Table[Length[Select[gug[Range[n]], SameQ@@Length/@Position[#, _Integer]&]], {n, 9}] PROG (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} b(n, k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v} seq(n)={my(M=Mat(vectorv(n, k, b(n, k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i, k]))} \\ Andrew Howroyd, Oct 26 2018 CROSSREFS Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312. Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179. Sequence in context: A173227 A099556 A057864 * A032273 A143522 A217389 Adjacent sequences:  A320151 A320152 A320153 * A320155 A320156 A320157 KEYWORD nonn AUTHOR Gus Wiseman, Oct 06 2018 EXTENSIONS Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018 STATUS approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)