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

%I A333005 #19 Jan 31 2025 07:26:37
%S A333005 1,6,135,5052,264270,17765100,1459311840,141655066560,15864853936680,
%T A333005 2013630348265200,285637924882787400,44782566595855149600,
%U A333005 7689608275439667376800,1435181273959520911824000,289287240571642427530416000,62630090604946453360419648000
%N A333005 Number of unrooted level-2 phylogenetic networks with n+1 labeled leaves, when multiple (i.e., parallel) edges are not allowed.
%H A333005 Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, <a href="http://user.math.uzh.ch/bouvel/publications/BouvelGambetteMansouri_Version2_WithoutMultipleEdges.mw">Maple worksheet</a>
%H A333005 Mathilde Bouvel, Philippe Gambette and Marefatollah Mansouri, <a href="https://arxiv.org/abs/1909.10460">Counting Phylogenetic Networks of level 1 and 2</a>, Version 3, arXiv:1909.10460 [math.CO], 2019.
%H A333005 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a333/A333005.java">Java program</a> (github)
%F A333005 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
%e A333005 a(3) = 135 is the number of unrooted level-2 phylogenetic networks with 4 labeled leaves.
%p A333005 # (See Links)
%p A333005 # second Maple program:
%p A333005 f:= z-> 1/(1-(3*z^5-16*z^4+32*z^3-30*z^2+12*z)/(4*(1-z)^4)):
%p A333005 a:= n-> n!*coeff(series(RootOf(U=z*f(U), U), z, n+1), z, n):
%p A333005 seq(a(n), n=1..23);  # _Alois P. Heinz_, Apr 01 2020
%t A333005 nmax = 16;
%t A333005 Module[{U, f, z},
%t A333005    U[_] = 0;
%t A333005    f[z_] := 1/(1 - (3*z^5 - 16*z^4 + 32*z^3 - 30*z^2 + 12*z)/(4*(1 - z)^4));
%t A333005    Do[U[z_] = z*f[U[z]] + O[z]^(nmax+1) // Normal, {nmax}];
%t A333005    Rest[CoefficientList[U[z], z]*Range[0, nmax]!]] (* _Jean-François Alcover_, Jan 31 2025 *)
%Y A333005 Cf. A328121, A328122, A328123, A328126, A333006.
%K A333005 nonn
%O A333005 1,2
%A A333005 _Mathilde Bouvel_, Mar 13 2020