A337901 The number of walks of length 2n+1 on the square lattice that start from the origin (0,0) and end at the vertex (3,0).

1, 25, 441, 7056, 108900, 1656369, 25050025, 378224704, 5712638724, 86394844900, 1308887012356, 19868414760000, 302198588499600, 4605510959127225, 70321771565375625, 1075697380745222400, 16483023079048102500, 252980753801047064100, 3888662839165553120100

Offset

1,2

Links

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

Formula

a(n) = A002054(n)^2.

G.f.: x*4F3(2,2,5/2,5/2; 1,4,4; 16*x).

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

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

Mathematica

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

Crossrefs

Cf. A002054.

Cf. A002894 (end at (0,0)), A060150 (end at (1,0)), A135389 (end at (1,1)), A337900 (at (2,0)), A337902 (at(2,1))

Sequence in context: A362428 A368161 A264382 * A260854 A001714 A016633

Adjacent sequences: A337898 A337899 A337900 * A337902 A337903 A337904

Keyword

nonn,easy,walk

Author

R. J. Mathar, Sep 29 2020

Status

approved