A306292 Number of asymmetric Dyck paths of semilength n.

0, 0, 2, 8, 32, 112, 394, 1360, 4736, 16544, 58324, 207088, 741184, 2671008, 9688410, 35344800, 129620480, 477590080, 1767170812, 6563935664, 24465914304, 91481858208, 343058261572, 1289901443168

Offset: 1

Author

Noah A Rosenberg, Feb 04 2019

Keywords

• nonn

Comments

An asymmetric Dyck path is a path that generates a distinct Dyck path when traversed in opposite order.

Examples

			For n=3, the a(2)=2 asymmetric Dyck paths are UUDDUD and UDUUDD.
		

Links

• 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.

Crossrefs

Equals twice A006079. Also A000108 minus A001405.

Programs

Mathematica

Table[Binomial[2 n, n]/(n + 1) - Binomial[n, Floor[n/2]], {n, 0, 30}]

PARI

a(n) = binomial(2*n, n)/(n+1) - binomial(n, n\2); \\ Michel Marcus, Jan 22 2020

Formula

a(n) = (2n)! / (n! (n+1))! - n! / ( (floor(n/2))! (ceiling(n/2))! ).