A333005 Number of unrooted level-2 phylogenetic networks with n+1 labeled leaves, when multiple (i.e., parallel) edges are not allowed.

1, 6, 135, 5052, 264270, 17765100, 1459311840, 141655066560, 15864853936680, 2013630348265200, 285637924882787400, 44782566595855149600, 7689608275439667376800, 1435181273959520911824000, 289287240571642427530416000, 62630090604946453360419648000

Offset

1,2

Links

Table of n, a(n) for n=1..16.

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

Mathematica

nmax = 16;
Module[{U, f, z},
   U[_] = 0;
   f[z_] := 1/(1 - (3*z^5 - 16*z^4 + 32*z^3 - 30*z^2 + 12*z)/(4*(1 - z)^4));
   Do[U[z_] = z*f[U[z]] + O[z]^(nmax+1) // Normal, {nmax}];
   Rest[CoefficientList[U[z], z]*Range[0, nmax]!]] (* Jean-François Alcover, Jan 31 2025 *)

Crossrefs

Cf. A328121, A328122, A328123, A328126, A333006.

Sequence in context: A096756 A356505 A241999 * A013299 A013295 A214132

Adjacent sequences: A333002 A333003 A333004 * A333006 A333007 A333008

Keyword

nonn

Author

Mathilde Bouvel, Mar 13 2020

Status

approved