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

Author

Mathilde Bouvel, Mar 13 2020

Keywords

• nonn

Examples

			a(3) = 135 is the number of unrooted level-2 phylogenetic networks with 4 labeled leaves.
		

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)

Crossrefs

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

Programs

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 *)

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