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 A320155 Number of series-reduced balanced rooted trees with n labeled leaves. 8
 1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 EXAMPLE The a(1) = 1 through a(5) = 11 rooted trees:   1  (12)  (123)    (1234)      (12345)                   ((12)(34))  ((12)(345))                   ((13)(24))  ((13)(245))                   ((14)(23))  ((14)(235))                               ((15)(234))                               ((23)(145))                               ((24)(135))                               ((25)(134))                               ((34)(125))                               ((35)(124))                               ((45)(123)) MATHEMATICA sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; phy2[labs_]:=If[Length[labs]==1, labs, Union@@Table[Sort/@Tuples[phy2/@ptn], {ptn, Select[sps[Sort[labs]], Length[#1]>1&]}]]; Table[Length[Select[phy2[Range[n]], SameQ@@Length/@Position[#, _Integer]&]], {n, 7}] 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; 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, A319312. Cf. A320154, A320160, A316624, A320169, A320173, A320179. Sequence in context: A278989 A000296 A032265 * A260320 A151273 A149271 Adjacent sequences:  A320152 A320153 A320154 * A320156 A320157 A320158 KEYWORD nonn AUTHOR Gus Wiseman, Oct 06 2018 EXTENSIONS Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018 STATUS approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)