A328121
Number of unrooted level-1 phylogenetic networks (also called galled trees) with (n+1) labeled leaves.
Original entry on oeis.org
1, 2, 15, 192, 3450, 79740, 2252880, 75227040, 2898481320, 126570502800, 6177380517000, 333231084648000, 19687828831070400, 1264341183311606400, 87691200344603856000, 6532556443068591936000, 520205544912884502672000, 44098092640676115673632000, 3964782594938523231457584000
Offset: 1
a(4) = 192 is the number of unrooted level-1 phylogenetic networks with 5 labeled leaves
A328122
Number of rooted level-1 phylogenetic networks with n labeled leaves.
Original entry on oeis.org
1, 3, 36, 723, 20280, 730755, 32171580, 1673573895, 100442870640, 6831585584775, 519288366989700, 43626178967384475, 4014060030471090600, 401443860155706643275, 43359414126089609047500, 5030039291029886037279375, 623762332206315636458124000, 82340968923184527676400655375, 11528273478697179256689693556500
Offset: 1
a(4) = 723 is the number of rooted level-1 phylogenetic networks with 4 labeled leaves.
- François Bienvenu, Jean-Jil Duchamps, Michael Fuchs, and Tsan-Cheng Yu, The B_2 index of galled trees, arXiv:2407.19454 [q-bio.PE], 2024. See p. 5.
- 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.
A328123
Number of unrooted level-2 phylogenetic networks with (n+1) labeled leaves, when multiple (i.e. parallel) edges are allowed.
Original entry on oeis.org
1, 9, 282, 14697, 1071750, 100467405, 11509922970, 1558302613245, 243426592473750, 43095781327975425, 8527098853816839450, 1864790504534293823025, 446647359698685492697350, 116281255808439040209815925, 32694665144001284972518220250
Offset: 1
a(3) = 282 is the number of unrooted level-2 phylogenetic networks with 4 labeled leaves.
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
a(3) = 135 is the number of unrooted level-2 phylogenetic networks with 4 labeled leaves.
-
# (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 *)
A333006
Number of rooted level-2 phylogenetic networks with n labeled leaves, when multiple (i.e., parallel) edges are not allowed.
Original entry on oeis.org
1, 18, 1143, 120078, 17643570, 3332111850, 769027554540, 209740414484160, 66001012966991340, 23537700706536311400, 9381525451337593738800, 4132780832455382525556600, 1993954501042287608709284400, 1045675186072945581517653088800
Offset: 1
a(3) = 1143 is the number of rooted level-2 phylogenetic networks with 3 labeled leaves.
-
# (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
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