A333006 Number of rooted level-2 phylogenetic networks with n labeled leaves, when multiple (i.e., parallel) edges are not allowed.

1, 18, 1143, 120078, 17643570, 3332111850, 769027554540, 209740414484160, 66001012966991340, 23537700706536311400, 9381525451337593738800, 4132780832455382525556600, 1993954501042287608709284400, 1045675186072945581517653088800

Offset: 1

Author

Mathilde Bouvel, Mar 13 2020

Keywords

• nonn

Examples

			a(3) = 1143 is the number of rooted level-2 phylogenetic networks with 3 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.

Crossrefs

Cf. A328121, A328122, A328123, A328126, A333005.

Programs

Maple

# (See Links)
# second Maple program:
f:= z-> 1/(1-(36*z-102*z^2+159*z^3-148*z^4+81*z^5-24*z^6+3*z^7)
         /(4*(1-z)^6)):
a:= n-> n!*coeff(series(RootOf(L=z*f(L), L), z, n+1), z, n):
seq(a(n), n=1..17);  # Alois P. Heinz, Apr 01 2020

Formula

E.g.f. satisfies L(z) = z*f(L(z)) where f(z) = 1 / (1 - (36*z-102*z^2+159*z^3-148*z^4+81*z^5-24*z^6+3*z^7)/(4*(1-z)^6)) [from Bouvel, Gambette, and Mansouri]. - Sean A. Irvine, Apr 01 2020