A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919

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

Formula

E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021

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