A320289 Number of phylogenetic trees with n labels and no singleton leaves.

0, 1, 1, 4, 11, 86, 477, 4810, 40679, 496522, 5662933, 81759910, 1169640551, 19622623190, 336215135973, 6455705990674, 128445712218263, 2785761076726066, 62980942321570981, 1525318051255683598, 38566041706375722071, 1032726237783455193662

Offset: 1

Author

Gus Wiseman, Oct 09 2018

Keywords

• nonn

Examples

			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))
		

Links

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

Crossrefs

Cf. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.

Programs

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]

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