OFFSET
4,1
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).
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
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.
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
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
KEYWORD
nonn,changed
AUTHOR
Noah A Rosenberg, Feb 14 2019
EXTENSIONS
a(4)=3 prepended by Noah A Rosenberg, Feb 10 2025
STATUS
approved