%I #23 Mar 26 2020 07:11:17
%S 1,1,8,1,16,29,1,32,95,80,1,64,313,426,195,1,128,1033,2320,1745,444,1,
%T 256,3411,12706,16347,6838,969,1,512,11265,69662,154259,112572,25897,
%U 2056,1,1024,37205,381964,1454495,1859660,752245,95292,4279
%N Square array T(n,k), n >= 1, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding walks in the n X k grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.
%e Square array T(n,k) begins:
%e 1, 1, 1, 1, 1, ...
%e 8, 16, 32, 64, 128, ...
%e 29, 95, 313, 1033, 3411, ...
%e 80, 426, 2320, 12706, 69662, ...
%e 195, 1745, 16347, 154259, 1454495, ...
%e 444, 6838, 112572, 1859660, 30549774, ...
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o import graphillion.tutorial as tl
%o def A(start, goal, n, k):
%o universe = tl.grid(n - 1, k - 1)
%o GraphSet.set_universe(universe)
%o paths = GraphSet.paths(start, goal)
%o return paths.len()
%o def A333509(n, k):
%o if n == 1: return 1
%o s = 0
%o for i in range(1, n + 1):
%o for j in range(k * n - n + 1, k * n + 1):
%o s += A(i, j, k, n)
%o return s
%o print([A333509(j + 1, i - j + 2) for i in range(9) for j in range(i + 1)])
%Y Columns k=2-3 give: A333510, A333511.
%Y Rows n=1-3 give: A000012, A000079(n+1), 2*A082574(n+1)+1.
%Y T(n,n) gives A121785(n-1).
%K nonn,tabl
%O 1,3
%A _Seiichi Manyama_, Mar 25 2020