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%I #8 Jan 02 2021 21:21:34
%S 0,1,1,1,2,2,4,5,9,14,24,39,69,116,205,357,632,1118,2001,3576,6445,
%T 11627,21080,38293,69819,127539,233644,428825,788832,1453589,2683602,
%U 4962167,9190155,17044522,31655676,58866237,109600849,204293047,381212823,712073862
%N Number of non-isomorphic phylogenetic trees with n nodes.
%C A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets. Each branching as well as each element of each leaf contributes to the number of nodes.
%H Andrew Howroyd, <a href="/A330627/b330627.txt">Table of n, a(n) for n = 1..500</a>
%F G.f.: A(x) satisfies A(x) = x*(1/(1-x) - A(x) - 2 + exp(Sum_{k>0} A(x^k)/k)). - _Andrew Howroyd_, Jan 02 2021
%e Non-isomorphic representatives of the a(2) = 1 through a(9) = 9 trees (commas and outer brackets elided):
%e 1 12 123 1234 12345 123456 1234567 12345678
%e (1)(2) (1)(23) (1)(234) (1)(2345) (1)(23456)
%e (12)(34) (12)(345) (12)(3456)
%e (1)(2)(3) (1)(2)(34) (123)(456)
%e (1)((2)(3)) (1)(2)(345)
%e (1)(23)(45)
%e (1)((2)(34))
%e (1)(2)(3)(4)
%e (12)((3)(4))
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, EulerT(v)[n] - v[n] + 1)); v} \\ _Andrew Howroyd_, Jan 02 2021
%Y Phylogenetic trees by number of labels are A005804, with unlabeled version A141268.
%Y Balanced phylogenetic trees are A320154.
%Y Cf. A000311, A000669, A001678, A004114, A005121, A007716, A048816, A060356, A330465, A330467, A330469.
%K nonn
%O 1,5
%A _Gus Wiseman_, Dec 28 2019
%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 02 2021
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