A260155 Number of walks of length 2n on the square lattice that start and end at (0,0) and avoid the negative quadrant.

1, 4, 32, 318, 3530, 41944, 522010, 6719018, 88726840, 1195527822, 16373466714, 227280520316, 3190715296368, 45226324937400, 646392346047930, 9305481272839662, 134815491199174476, 1964195875748858812, 28761433275110249932, 423052415434610432816

Offset: 0

Author

Mireille Bousquet-Mélou, Nov 09 2015

Keywords

• nonn
• easy
• walk

Examples

			When n=1 the four walks are NS, EW, SN, WE.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800
• M. Bousquet-Mélou, Plane lattice walks avoiding a quadrant, arXiv:1511.02111 [math.CO], 2015.

Crossrefs

Cf. A060898 for walks starting from (0,0) but in which the final point is not prescribed.

Programs

Mathematica

f[x_, n_] := x Pochhammer[x+1, n-1];
a[n_] := 4 16^n/3^5 (3^4 f[1/2, n] f[1/2, n + 1]/(f[2, n] f[2, n + 1]) + 4 (24n^2 + 60n + 29) f[1/2, n] f[7/6, n]/(f[2, n + 1] f[4/3, n + 1]) - 2 (12n^2 + 30n + 5) f[1/2, n] f[5/6, n]/(f[2, n + 1] f[5/3, n + 1]));
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 25 2018 *)

Formula

a(n) = 4*16^n/ 3^5 * ( 3^4 *f(1/2,n)* f(1/2,n+1)/ (f(2,n) * f(2,n+1)) + 4 *(24*n^2+60*n +29)* f(1/2,n)* f(7/6,n) /(f(2,n+1) *f(4/3, n+1)) -2 *(12*n^2+30*n+5) * f(1/2,n)*f(5/6,n) /(f(2,n+1)*f(5/3,n+1)) ) where f(m,n) is the ascending factorial m*(m+1)*...*(m+n-1) (proved).