A145600 a(n) is the number of walks from (0,0) to (0,1) that remain in the upper half-plane y >= 0 using (2*n - 1) unit steps either up (U), down (D), left (L) or right (R).

1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700

Offset: 1

Author

Peter Bala, Oct 14 2008

Keywords

• easy
• nonn

Comments

Cf. A000891, which enumerates walks in the upper half-plane starting and finishing at the origin. See also A145601, A145602 and A145603. This sequence is the central column taken from triangle A145596, which enumerates walks in the upper half-plane starting at the origin and finishing on the horizontal line y = 1.

Examples

			a(2) = 8: the 8 walks from (0,0) to (0,1) of three steps are
UDU, UUD, URL, ULR, RLU, LRU, RUL and LUR.
		

References

• M. Dukes and Y. Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013, Pages 816-842. - From N. J. A. Sloane, Feb 21 2013

Links

• R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Crossrefs

Cf. A000891, A145596, A145601, A145602, A145603.

Programs

Maple

a(n) := 1/n*binomial(2*n,n+1)*binomial(2*n,n-1);
seq(a(n),n = 1..19);

Formula

a(n) = 1/n*binomial(2*n,n+1)*binomial(2*n,n-1).

a(n) = A135389(n-1)/(n+1). - R. J. Mathar, Jul 14 2013

D-finite with recurrence (n+1)^2*a(n) -4*n*(5*n-1)*a(n-1) +16*(2*n-3)^2*a(n-2)=0. - R. J. Mathar, Jul 14 2013