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A333005
Number of unrooted level-2 phylogenetic networks with n+1 labeled leaves, when multiple (i.e., parallel) edges are not allowed.
3
1, 6, 135, 5052, 264270, 17765100, 1459311840, 141655066560, 15864853936680, 2013630348265200, 285637924882787400, 44782566595855149600, 7689608275439667376800, 1435181273959520911824000, 289287240571642427530416000, 62630090604946453360419648000
OFFSET
1,2
LINKS
Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, Maple worksheet
Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, Counting Phylogenetic Networks of level 1 and 2, Version 3, arXiv:1909.10460 [math.CO], 2019.
Sean A. Irvine, Java program (github)
FORMULA
E.g.f. satisfies U(z) = z*f(U(z)) where f(z) = 1 / (1 - (3*z^5-16*z^4+32*z^3-30*z^2+12*z)/(4*(1-z)^4)) [from Bouvel, Gambette, and Mansouri]. - Sean A. Irvine, Apr 01 2020
EXAMPLE
a(3) = 135 is the number of unrooted level-2 phylogenetic networks with 4 labeled leaves.
MAPLE
# (See Links)
# second Maple program:
f:= z-> 1/(1-(3*z^5-16*z^4+32*z^3-30*z^2+12*z)/(4*(1-z)^4)):
a:= n-> n!*coeff(series(RootOf(U=z*f(U), U), z, n+1), z, n):
seq(a(n), n=1..23); # Alois P. Heinz, Apr 01 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Mathilde Bouvel, Mar 13 2020
STATUS
approved