1,2

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.

Mark Dukes and Yvan Le Borgne, Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial, Journal of Combinatorial Theory, Series A, Vol. 120, No. 4 (May 2013), pages 816-842. - From N. J. A. Sloane, Feb 21 2013

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

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

a(n) ~ 16^n / (Pi * n^2). - Amiram Eldar, Oct 08 2025

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.

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

seq(a(n), n = 1..19);

a[n_] := Binomial[2*n, n+1] * Binomial[2*n, n-1] / n; Array[a, 20] (* Amiram Eldar, Oct 08 2025 *)

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

Sequence in context: A015578 A386868 A268085 * A231617 A094735 A383951

Adjacent sequences: A145597 A145598 A145599 * A145601 A145602 A145603

easy,nonn

Peter Bala, Oct 14 2008

approved