cp's OEIS Frontend

A307697 Number of Evolutionary Duplication-Loss-histories with n leaves of the caterpillar species tree with 3 leaves.

%I A307697 #22 Jul 31 2019 11:13:15
%S A307697 3,19,159,1565,17022,197928,2413494,30490089,395828145,5250493688,
%T A307697 70863932052,970121212741,13439019867456,188038364992270,
%U A307697 2653560128625570,37723174042204665,539726553801797610,7765849268430279390
%N A307697 Number of Evolutionary Duplication-Loss-histories with n leaves of the caterpillar species tree with 3 leaves.
%C A307697 An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.
%H A307697 C. Chauve, Y. Ponty, M. Wallner, <a href="https://arxiv.org/abs/1905.04971">Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models</a>, arXiv preprint arXiv:1905.04971 [math-CO], 2019.
%F A307697 G.f.: 1/2 - (1/2)*sqrt(-4 - t*u + 3*t + 3*u) where t = sqrt(1 - 4*z) and u = sqrt(-5 + 6*t + 4*z).
%e A307697 The caterpillar species tree with 3 leaves is equal to
%e A307697       a
%e A307697      / \
%e A307697     b   3
%e A307697    / \
%e A307697   1   2
%e A307697 For convenience the internal nodes are labeled by a,b, and the leaves by 1,2,3. The associated nodes in the histories will be denoted by the same labels.
%e A307697 The a(1)=3 histories with n=1 leaf are created by the following growth process:
%e A307697       a     a     a
%e A307697      /     /       \
%e A307697     b     b         3
%e A307697    /       \
%e A307697   1         2
%e A307697 after two loss events each.
%Y A307697 Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307698 (4 leaves), A307700 (5 leaves).
%K A307697 nonn
%O A307697 1,1
%A A307697 _Michael Wallner_, Apr 22 2019