This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306423 #42 Feb 12 2025 20:34:17
%S A306423 3,11,37,124,420,1441,5005,17576,62322,222870,802978,2912168,10623470,
%T A306423 38956365,143521725,530985360,1971965490,7348812570,27472909590,
%U A306423 103002205800,387205269360,1459146890058,5511120747282,20858962792624,79103096214100
%N A306423 Number of coalescent histories for pseudocaterpillar gene trees G and caterpillar species trees S.
%C A306423 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).
%C A306423 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
%H A306423 E. Alimpiev and N. A. Rosenberg, <a href="https://doi.org/10.1016/j.aam.2021.102265">Enumeration of coalescent histories for caterpillar species trees and p-pseudocaterpillar gene trees</a>, Adv. Appl. Math. 131 (2021), 102265.
%H A306423 N. A. Rosenberg and J. H. Degnan, <a href="https://doi.org/10.1016/j.tpb.2009.12.004">Coalescent histories for discordant gene trees and species trees</a>. Theor. Pop. Biol. 77 (2010), 145-151.
%F A306423 a(n) = (19*n-40)*(n-3)*(2*n-2)!/(4*n!*(n-1)!*(2*n-3)*(2*n-5)).
%F A306423 a(n) = (19*n-40)*(n-3)*C(n-1)/((2*n-3)*(2*n-5)), where C(n) is the Catalan numbers A000108.
%F A306423 G.f.: ((2 - 7*x + x^2) +(-2 + 3*x + x^2)*sqrt(1-4*x))/2. - _G. C. Greubel_, Mar 07 2019
%F A306423 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
%F A306423 a(n) ~ 19 * 2^(2*n - 6) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 15 2022
%e A306423 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).
%t A306423 Table[(19n-40)(n-3) Binomial[2n-2, n-1]/(4n(2n-3)(2n-5)), {n, 5, 30}]
%o A306423 (PARI) {a(n)=(19*n-40)*(n-3)*binomial(2*n-2, n-1)/(4*n*(2*n-3)*(2*n-5))};
%o A306423 for(n=5,30, print1(a(n), ", ")) \\ _G. C. Greubel_, Mar 07 2019
%o A306423 (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
%o A306423 (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
%o A306423 (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
%Y A306423 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.
%K A306423 nonn
%O A306423 4,1
%A A306423 _Noah A Rosenberg_, Feb 14 2019
%E A306423 a(4)=3 prepended by _Noah A Rosenberg_, Feb 10 2025