A307942 Number of evolutionary duplication-loss-histories of the complete binary species tree with 8 leaves.

8, 148, 3376, 89390, 2624872, 82866636, 2755019736, 95135709027, 3380416782760, 122798718575216, 4539685792433848, 170225552910292438, 6458330316575589176, 247456381334355675220, 9561546562984390785960, 372141845574597078971490, 14575950501012888889866120

Offset: 1

Author

Cedric Chauve, May 07 2019

Keywords

• nonn

Comments

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 complete binary species tree S of size k is a complete binary tree with k leaves. 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.

Examples

			See A307941 (complete binary species tree with 4 leaves).
		

Crossrefs

Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).

Cf. A307696, A307697, A307698, A307700 (caterpillar species tree with 2, 3, 4, 5 leaves).

Programs

PARI

z='z+O('z^20); my(t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(1+6*u-6*t-4*z)); Vec(1/2-(1/2)*sqrt(-5+6*v-6*u+6*t+4*z)) \\ Jianing Song, Jul 29 2019

Formula

G.f.: 1/2-(1/2)*sqrt(-5+6*v-6*u+6*t+4*z) where t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z) and v = sqrt(1+6*u-6*t-4*z).