A135389 - OEIS

%I #15 Sep 29 2020 07:45:58

%S 2,24,300,3920,52920,731808,10306296,147232800,2127513960,31031617760,

%T 456164781072,6749962774464,100445874620000,1502052155856000,

%U 22557604697766000,340044833169460800,5143178101688094600

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

%C 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).

%H S. Hollos and R. Hollos, <a href="http://www.exstrom.com/math/lattice/latpath.html">Lattice Paths and Walks</a>.

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

%F G.f.: 2*2F1(3/2,3/2; 3; 16*x). - _Mark van Hoeij_, Apr 06 2013

%F D-finite with recurrence n*(n+2)*a(n) -4*(2*n+1)^2*a(n-1)=0. - _R. J. Mathar_, Jul 14 2013

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

%e G.f. = 2 + 24*x + 300*x^2 + 3920*x^3 + 731808*x^4 + 10306296*x^5 + ... - _Michael Somos_, Oct 17 2019

%p series( 2*hypergeom([3/2, 3/2],[3],16*x), x=0, 20);  # _Mark van Hoeij_, Apr 06 2013

%t Table[Binomial[2n + 2, n] Binomial[2n + 2, n + 1], {n, 0, 19}] (* _Alonso del Arte_, Apr 06 2013 *)

%Y Cf. A002894, A060150, A337900-A337902.

%K easy,nonn,walk

%O 0,1

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