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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 28 2019
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 02 2021
STATUS
approved