OFFSET
0,2
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.
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).
EXAMPLE
When n=1 the four walks are NS, EW, SN, WE.
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 *)
CROSSREFS
KEYWORD
nonn,easy,walk
AUTHOR
Mireille Bousquet-Mélou, Nov 09 2015
STATUS
approved