A333006 -id:A333006 - cp's OEIS frontend

A328121 Number of unrooted level-1 phylogenetic networks (also called galled trees) with (n+1) labeled leaves.

1, 2, 15, 192, 3450, 79740, 2252880, 75227040, 2898481320, 126570502800, 6177380517000, 333231084648000, 19687828831070400, 1264341183311606400, 87691200344603856000, 6532556443068591936000, 520205544912884502672000, 44098092640676115673632000, 3964782594938523231457584000

Offset: 1

Mathilde Bouvel, Oct 04 2019

nonn

			a(4) = 192 is the number of unrooted level-1 phylogenetic networks with 5 labeled leaves

Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, Maple worksheet
Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, Counting Phylogenetic Networks of level 1 and 2, arXiv:1909.10460 [math.CO], 2019.
Charles Semple and Mike Steel, Unicyclic networks: compatibility and enumeration, Transactions on Computational Biology and Bioinformatics (3:1 pp.398--401), 2006.

Cf. A328122, A328123, A328126, A333005, A333006.

Maple
```
# see links section
```

Semple and Steele provide a summation formula for a(n) (see their Theorem 4).

Bouvel, Gambette and Mansouri provide (among other additional results) an equation for the associated exponential generating function, and an asymptotic estimate of a(n). See their Section 4.

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

Original entry on oeis.org

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

Offset: 1

Mathilde Bouvel, Mar 13 2020

nonn

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

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)

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

# (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

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 *)

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

cp's OEIS Frontend

A328121 Number of unrooted level-1 phylogenetic networks (also called galled trees) with (n+1) labeled leaves.

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