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

3, 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: 4

Author

Noah A Rosenberg, Feb 14 2019

Keywords

• nonn

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).

A slightly different definition of pseudocaterpillar trees applies for n=4 and adds term a(4)=3 to a (see Table 4 of Alimpiev & Rosenberg 2021). - Noah A Rosenberg, Feb 10 2025

Examples

			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).
		

Links

• 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.

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.

Programs

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

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

Mathematica

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

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

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

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) +(-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

Extensions

a(4)=3 prepended by Noah A Rosenberg, Feb 10 2025