A337902 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (2,1).

3, 50, 735, 10584, 152460, 2208492, 32207175, 472780880, 6982113996, 103673813880, 1546866469148, 23179817220000, 348690679038000, 5263441096145400, 79698007774092375, 1210159553338375200, 18422202264818467500, 281089726445607849000, 4297995769604032395900

Offset

1,1

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = binomial(2*n+1,n-1)*binomial(2*n+1,n) = A002054(n)*A001700(n).

G.f.: 3*x*3F2(2,5/2,5/2; 3,4; 16*x).

D-finite with recurrence (n-1)*(n+2)*(n+1)*a(n) - 4*n*(2*n+1)^2*a(n-1) = 0.

A135389(n) = 2*A060150(n+1) +2*a(n).

a(n) ~ 4^(2*n+1) / (Pi*n). - Amiram Eldar, Oct 16 2025

Example

a(1) = 3 represents 3 walks of length 3: RRU, URR and RUR.

Mathematica

a[n_] := Binomial[2*n+1, n-1] * Binomial[2*n+1, n]; Array[a, 20] (* Amiram Eldar, Oct 16 2025 *)

Crossrefs

Cf. A002054, A001700.

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337900 (at (2,0)), A337901 (at (3,0))

Sequence in context: A034201 A099346 A075184 * A246283 A326250 A396808

Adjacent sequences: A337899 A337900 A337901 * A337903 A337904 A337905

Keyword

nonn,easy,walk

Author

R. J. Mathar, Sep 29 2020

Status

approved