A306423 Number of coalescent histories for pseudocaterpillar gene trees G and caterpillar species trees S.

11, 37, 124, 420, 1441, 5005, 17576, 62322, 222870, 802978, 2912168, 10623470, 38956365, 143521725, 530985360, 1971965490, 7348812570, 27472909590, 103002205800, 387205269360, 1459146890058, 5511120747282, 20858962792624, 79103096214100

Offset

5,1

Comments

Consider a binary, rooted, leaf-labeled caterpillar species tree G = (...((((A_1, A_2), A_3), A_4), A_5),..., A_n) and a binary, rooted, leaf-labeled pseudocaterpillar gene tree (...(((A_1, A_2), (A_3, A_4)), A_5),..., A_n). The pseudocaterpillar family of trees is defined for n>=5 leaves (Rosenberg 2007). Sequence a(n) gives the number of coalescent histories for (G,S).

Links

Table of n, a(n) for n=5..28.

E. Alimpiev and N. A. Rosenberg, Enumeration of coalescent histories for caterpillar species trees and p-pseudocaterpillar gene trees, Adv. Appl. Math. 131 (2021), 102265.

N. A. Rosenberg and J. H. Degnan, Coalescent histories for discordant gene trees and species trees. Theor. Pop. Biol. 77 (2010), 145-151.

Formula

a(n) = (19*n-40)*(n-3)*(2*n-2)!/(4*n!*(n-1)!*(2*n-3)*(2*n-5)).

a(n) = (19*n-40)*(n-3)*C(n-1)/((2*n-3)*(2*n-5)), where C(n) is the Catalan numbers A000108.

G.f.: ((2 - 7*x + x^2 - 6*x^4) +(-2 + 3*x + x^2)*sqrt(1-4*x))/2. - G. C. Greubel, Mar 07 2019

D-finite with recurrence: +2*n*a(n) +(-11*n+18)*a(n-1) +(11*n-38)*a(n-2) +2*(2*n-11)*a(n-3)=0. - R. J. Mathar, Jan 27 2020

a(n) ~ 19 * 2^(2*n - 6) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 15 2022

Example

For n=5, consider species tree ((((A_1, A_2), A_3), A_4), A_5) and gene tree ((((A_1, A_2), (A_3, A_4)), A_5). Label the nodes of the species tree 1, 2, 3, 4, from the cherry to the root, identifying each node with its immediate ancestral edge. Annotate the coalescent histories by vectors whose entries, in order, denote the locations of the coalescences of (A_1, A_2), (A_3, A_4), ((A_1, A_2), (A_3, A_4)), and ((((A_1, A_2), (A_3, A_4)), A_5). The a(5)=11 coalescent histories are (1,3,3,4), (1,3,4,4), (1,4,4,4), (2,3,3,4), (2,3,4,4), (2,4,4,4), (3,3,3,4), (3,3,4,4), (3,4,4,4), (4,3,4,4), and (4,4,4,4).

Mathematica

Table[(19n-40)(n-3) Binomial[2n-2, n-1]/(4n(2n-3)(2n-5)), {n, 5, 30}]

Prog

(PARI) {a(n)=(19*n-40)*(n-3)*binomial(2*n-2, n-1)/(4*n*(2*n-3)*(2*n-5))};
for(n=5, 30, print1(a(n), ", ")) \\ G. C. Greubel, Mar 07 2019
(Magma) [(19*n-40)*(n-3)*Binomial(2*n-2, n-1)/(4*n*(2*n-3)*(2*n-5)): n in [5..30]]; // G. C. Greubel, Mar 07 2019
(Sage) [(19*n-40)*(n-3)*binomial(2*n-2, n-1)/(4*n*(2*n-3)*(2*n-5)) for n in (5..30)] # G. C. Greubel, Mar 07 2019
(GAP) List([5..30], n-> (19*n-40)*(n-3)*Binomial(2*n-2, n-1)/(4*n*(2*n-3)*(2*n-5))) # G. C. Greubel, Mar 07 2019

Crossrefs

A000108 gives the number of coalescent histories for matching caterpillar gene trees and species trees. A070031 gives the number of coalescent histories for matching pseudocaterpillar gene trees and species trees.

Sequence in context: A210321 A356278 A355630 * A287018 A152094 A227412

Adjacent sequences: A306420 A306421 A306422 * A306424 A306425 A306426

Keyword

nonn

Author

Noah A Rosenberg, Feb 14 2019

Status

approved