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 A306292 Number of asymmetric Dyck paths of semilength n. 0
 0, 0, 2, 8, 32, 112, 394, 1360, 4736, 16544, 58324, 207088, 741184, 2671008, 9688410, 35344800, 129620480, 477590080, 1767170812, 6563935664, 24465914304, 91481858208, 343058261572, 1289901443168 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS An asymmetric Dyck path is a path that generates a distinct Dyck path when traversed in opposite order. LINKS Table of n, a(n) for n=1..24. J. Bonin, A. de Mier, and M. Noy, Lattice path matroids: enumerative aspects and Tutte polynomials, J Comb Theory Ser A 104 (2003), 63-94. L.-H. Deng, E. Y. P. Deng, and L. W. Shapiro, The Riordan group and symmetric lattice paths, arXiv:0906.1844 [math.CO] (2009). Zoe M. Himwich and Noah A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. FORMULA a(n) = (2n)! / (n! (n+1))! - n! / ( (floor(n/2))! (ceiling(n/2))! ). EXAMPLE For n=3, the a(2)=2 asymmetric Dyck paths are UUDDUD and UDUUDD. MATHEMATICA Table[Binomial[2 n, n]/(n + 1) - Binomial[n, Floor[n/2]], {n, 0, 30}] PROG (PARI) a(n) = binomial(2*n, n)/(n+1) - binomial(n, n\2); \\ Michel Marcus, Jan 22 2020 CROSSREFS Equals twice A006079. Also A000108 minus A001405. Sequence in context: A227971 A267661 A052481 * A067897 A145682 A363109 Adjacent sequences: A306289 A306290 A306291 * A306293 A306294 A306295 KEYWORD nonn AUTHOR Noah A Rosenberg, Feb 04 2019 STATUS approved

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Last modified June 22 02:28 EDT 2024. Contains 373561 sequences. (Running on oeis4.)