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 A306423 Number of coalescent histories for pseudocaterpillar gene trees G and caterpillar species trees S. 0
 11, 37, 124, 420, 1441, 5005, 17576, 62322, 222870, 802978, 2912168, 10623470, 38956365, 143521725, 530985360, 1971965490, 7348812570, 27472909590, 103002205800, 387205269360, 1459146890058, 5511120747282, 20858962792624, 79103096214100 (list; graph; refs; listen; history; text; internal format)
 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 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 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: A229612 A297797 A210321 * A287018 A152094 A227412 Adjacent sequences:  A306420 A306421 A306422 * A306424 A306425 A306426 KEYWORD nonn AUTHOR Noah A Rosenberg, Feb 14 2019 STATUS approved

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Last modified July 19 08:48 EDT 2019. Contains 325155 sequences. (Running on oeis4.)