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

%I A320289 #8 Oct 27 2018 01:07:41
%S A320289 0,1,1,4,11,86,477,4810,40679,496522,5662933,81759910,1169640551,
%T A320289 19622623190,336215135973,6455705990674,128445712218263,
%U A320289 2785761076726066,62980942321570981,1525318051255683598,38566041706375722071,1032726237783455193662
%N A320289 Number of phylogenetic trees with n labels and no singleton leaves.
%H A320289 Andrew Howroyd, <a href="/A320289/b320289.txt">Table of n, a(n) for n = 1..200</a>
%e A320289 The a(2) = 1 through a(5) = 11 phylogenetic trees:
%e A320289   (12)  (123)  (1234)      (12345)
%e A320289                ((12)(34))  ((12)(345))
%e A320289                ((13)(24))  ((13)(245))
%e A320289                ((14)(23))  ((14)(235))
%e A320289                            ((15)(234))
%e A320289                            ((23)(145))
%e A320289                            ((24)(135))
%e A320289                            ((25)(134))
%e A320289                            ((34)(125))
%e A320289                            ((35)(124))
%e A320289                            ((45)(123))
%t A320289 numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
%t A320289 rotf[n_]:=rotf[n]=If[n==1,0,1+Sum[numSetPtnsOfType[p]*Times@@rotf/@p,{p,Select[IntegerPartitions[n],Length[#]>1&]}]];
%t A320289 Array[rotf,20]
%o A320289 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320289 b(n,k)={my(v=vector(n)); for(n=2, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
%o A320289 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
%Y A320289 Cf. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.
%K A320289 nonn
%O A320289 1,4
%A A320289 _Gus Wiseman_, Oct 09 2018