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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..26.

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).

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.

EXAMPLE

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.

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}]

CROSSREFS

A000108 minus A005817.

Sequence in context: A063782 A071718 A261058 * A134952 A184436 A149028

Adjacent sequences:  A306292 A306293 A306294 * A306296 A306297 A306298

KEYWORD

nonn

AUTHOR

Noah A Rosenberg, Feb 04 2019

STATUS

approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)