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A320289 Number of phylogenetic trees with n labels and no singleton leaves. 5
0, 1, 1, 4, 11, 86, 477, 4810, 40679, 496522, 5662933, 81759910, 1169640551, 19622623190, 336215135973, 6455705990674, 128445712218263, 2785761076726066, 62980942321570981, 1525318051255683598, 38566041706375722071, 1032726237783455193662 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

EXAMPLE

The a(2) = 1 through a(5) = 11 phylogenetic trees:

  (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

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];

rotf[n_]:=rotf[n]=If[n==1, 0, 1+Sum[numSetPtnsOfType[p]*Times@@rotf/@p, {p, Select[IntegerPartitions[n], Length[#]>1&]}]];

Array[rotf, 20]

PROG

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

b(n, k)={my(v=vector(n)); for(n=2, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); 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. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.

Sequence in context: A181275 A000850 A004796 * A298858 A181267 A280036

Adjacent sequences:  A320286 A320287 A320288 * A320290 A320291 A320292

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 09 2018

STATUS

approved

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Last modified April 13 03:43 EDT 2021. Contains 342934 sequences. (Running on oeis4.)