A333685 - OEIS

%I #23 Apr 03 2020 01:31:17

%S 1,1,10,1,101,7056,1,1105,610765,462755440,1,12046,53968755,

%T 365962179700,2593165016903538,1,131399,4775133828,294346514811753,

%U 18855848354902159112,1243982213040307428318660

%N Triangle T(n,k), n>=0, 0 <= k <= n, read by rows, where T(n,k) is the number of self-avoiding paths in (2*n+1) X (2*k+1) grid starting the upper left corner, passing through the center of grid and finishing the lower right corner.

%F T(n,k) = T(k,n).

%e Triangle starts:

%e ====================================================================

%e n\k| 0        1             2                   3                 4

%e ---|----------------------------------------------------------------

%e 0  | 1;

%e 1  | 1,      10;

%e 2  | 1,     101,         7056;

%e 3  | 1,    1105,       610765,          462755440;

%e 4  | 1,   12046,     53968755,       365962179700, 2593165016903538;

%e 5  | 1,  131399,   4775133828,    294346514811753, ...

%e 6  | 1, 1433341, 422813081886, 237970057189444731, ...

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o import graphillion.tutorial as tl

%o def A333685(n, k):

%o     if n == 0 or k == 0: return 1

%o     universe = tl.grid(2 * n, 2 * k)

%o     GraphSet.set_universe(universe)

%o     start, goal = 1, (2 * n + 1) * (2 * k + 1)

%o     paths = GraphSet.paths(start, goal).including((start + goal) // 2)

%o     return paths.len()

%o print([A333685(n, k) for n in range(6) for k in range(n + 1)])

%Y Columns k=0..2 give A000012, A333686, A333689.

%Y T(n,n) gives A121787.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Apr 02 2020