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 A330627 Number of non-isomorphic phylogenetic trees with n nodes. 4
 0, 1, 1, 1, 2, 2, 4, 5, 9, 14, 24, 39, 69, 116, 205, 357, 632, 1118, 2001, 3576, 6445, 11627, 21080, 38293, 69819, 127539, 233644, 428825, 788832, 1453589, 2683602, 4962167, 9190155, 17044522, 31655676, 58866237, 109600849, 204293047, 381212823, 712073862 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..500 FORMULA 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 EXAMPLE Non-isomorphic representatives of the a(2) = 1 through a(9) = 9 trees (commas and outer brackets elided):   1  12  123  1234    12345    123456     1234567      12345678               (1)(2)  (1)(23)  (1)(234)   (1)(2345)    (1)(23456)                                (12)(34)   (12)(345)    (12)(3456)                                (1)(2)(3)  (1)(2)(34)   (123)(456)                                           (1)((2)(3))  (1)(2)(345)                                                        (1)(23)(45)                                                        (1)((2)(34))                                                        (1)(2)(3)(4)                                                        (12)((3)(4)) PROG (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} 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 CROSSREFS Phylogenetic trees by number of labels are A005804, with unlabeled version A141268. Balanced phylogenetic trees are A320154. Cf. A000311, A000669, A001678, A004114, A005121, A007716, A048816, A060356, A330465, A330467, A330469. Sequence in context: A007147 A230380 A127968 * A188541 A037026 A116651 Adjacent sequences:  A330624 A330625 A330626 * A330628 A330629 A330630 KEYWORD nonn AUTHOR Gus Wiseman, Dec 28 2019 EXTENSIONS Terms a(11) and beyond from Andrew Howroyd, Jan 02 2021 STATUS approved

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Last modified June 21 02:43 EDT 2021. Contains 345351 sequences. (Running on oeis4.)