A135389 Number of walks of length 2*n+2 from origin to (1,1) in a square lattice.

2, 24, 300, 3920, 52920, 731808, 10306296, 147232800, 2127513960, 31031617760, 456164781072, 6749962774464, 100445874620000, 1502052155856000, 22557604697766000, 340044833169460800, 5143178101688094600

Offset

0,1

Comments

a(n) is the number of walks of length 2n+2 in an infinite square lattice that begin at the origin and end at (1,1) using steps (1,0), (-1,0), (0,1), (0,-1).

Links

Table of n, a(n) for n=0..16.

S. Hollos and R. Hollos, Lattice Paths and Walks.

Formula

a(n) = binomial(2n+2,n) * binomial(2n+2,n+1) = A001791(n+1)*A000984(n+1).

G.f.: 2*2F1(3/2,3/2; 3; 16*x). - Mark van Hoeij, Apr 06 2013

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

E.g.f.: Sum_{n>0} a(n-1) * x^(2*n)/(2*n)! = BesselI(1, 2*x)^2. - Michael Somos, Oct 17 2019

Example

G.f. = 2 + 24*x + 300*x^2 + 3920*x^3 + 731808*x^4 + 10306296*x^5 + ... - Michael Somos, Oct 17 2019

Maple

series( 2*hypergeom([3/2, 3/2], [3], 16*x), x=0, 20);  # Mark van Hoeij, Apr 06 2013

Mathematica

Table[Binomial[2n + 2, n] Binomial[2n + 2, n + 1], {n, 0, 19}] (* Alonso del Arte, Apr 06 2013 *)

Crossrefs

Cf. A002894, A060150, A337900-A337902.

Sequence in context: A355951 A065101 A052739 * A336310 A065513 A246190

Adjacent sequences: A135386 A135387 A135388 * A135390 A135391 A135392

Keyword

easy,nonn,walk

Author

Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007

Status

approved