A333685 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.

1, 1, 10, 1, 101, 7056, 1, 1105, 610765, 462755440, 1, 12046, 53968755, 365962179700, 2593165016903538, 1, 131399, 4775133828, 294346514811753, 18855848354902159112, 1243982213040307428318660

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

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

Example

Triangle starts:
====================================================================
n\k| 0        1             2                   3                 4
---|----------------------------------------------------------------
0  | 1;
1  | 1,      10;
2  | 1,     101,         7056;
3  | 1,    1105,       610765,          462755440;
4  | 1,   12046,     53968755,       365962179700, 2593165016903538;
5  | 1,  131399,   4775133828,    294346514811753, ...
6  | 1, 1433341, 422813081886, 237970057189444731, ...

Prog

(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333685(n, k):
    if n == 0 or k == 0: return 1
    universe = tl.grid(2 * n, 2 * k)
    GraphSet.set_universe(universe)
    start, goal = 1, (2 * n + 1) * (2 * k + 1)
    paths = GraphSet.paths(start, goal).including((start + goal) // 2)
    return paths.len()
print([A333685(n, k) for n in range(6) for k in range(n + 1)])

Crossrefs

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

T(n,n) gives A121787.

Sequence in context: A178870 A075505 A130310 * A288050 A288503 A051523

Adjacent sequences: A333682 A333683 A333684 * A333686 A333687 A333688

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 02 2020

Status

approved