A306295 Maximal number of coalescent histories among non-matching pairs consisting of a caterpillar gene tree and a caterpillar species tree with n+2 leaves.

1, 3, 10, 32, 107, 359, 1234, 4274, 15032, 53242, 190588, 686272, 2490399, 9081375, 33312770, 122692130, 453999656, 1685601038, 6282014804, 23478897364, 88026769844, 330831420218, 1246635155180, 4707414286652, 17815452662152, 67546709440004, 256595322436760

Offset: 1

Author

Noah A Rosenberg, Feb 04 2019

Keywords

• nonn

Examples

			For n=1, a non-matching caterpillar gene tree and species tree with n+2=3 leaves have only one coalescent history: all coalescences must take place above the root of the species tree. Hence, a(1)=1.
		

Links

• Z. M. Himwich and N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [q-bio.pE] (2019); Adv. Appl. Math. 113 (2020), 101939.

Crossrefs

A000108 minus A005817.

Programs

Mathematica

b[n_] :=
Binomial[2 n - 2, n - 1]/
   n - (2 Floor[(n - 1)/2])!/(Floor[(n - 1)/2]! Floor[(n + 1)/
         2]!) (2 Ceiling[(n - 1)/2])!/(Ceiling[(n - 1)/
         2]! Ceiling[(n + 1)/2]!)
a[n_] := b[n+2]
Table[a[n], {n,1,30}]

Formula

a(n) = C(n+1) - C(floor((n+1)/2))*C(ceiling((n+1)/2)), where C(n) is the n-th term in the Catalan sequence A000108.