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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, 256595322436760
OFFSET
1,2
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.
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
Sequence in context: A063782 A071718 A261058 * A356499 A134952 A184436
KEYWORD
nonn
AUTHOR
Noah A Rosenberg, Feb 04 2019
STATUS
approved