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A260155
Number of walks of length 2n on the square lattice that start and end at (0,0) and avoid the negative quadrant.
1
1, 4, 32, 318, 3530, 41944, 522010, 6719018, 88726840, 1195527822, 16373466714, 227280520316, 3190715296368, 45226324937400, 646392346047930, 9305481272839662, 134815491199174476, 1964195875748858812, 28761433275110249932, 423052415434610432816
OFFSET
0,2
LINKS
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
Cf. A060898 for walks starting from (0,0) but in which the final point is not prescribed.
Sequence in context: A197715 A369026 A237019 * A047734 A220118 A060174
KEYWORD
nonn,easy,walk
AUTHOR
STATUS
approved