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A333580 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in an n X k grid starting at the lower left corner and finishing in the upper right corner.

%I A333580 #27 Jan 31 2022 15:51:21
%S A333580 1,1,1,1,0,1,1,1,1,1,1,0,2,0,1,1,1,4,4,1,1,1,0,8,0,8,0,1,1,1,16,20,20,
%T A333580 16,1,1,1,0,32,0,104,0,32,0,1,1,1,64,111,378,378,111,64,1,1,1,0,128,0,
%U A333580 1670,0,1670,0,128,0,1,1,1,256,624,6706,10204,10204,6706,624,256,1,1
%N A333580 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in an n X k grid starting at the lower left corner and finishing in the upper right corner.
%H A333580 Andrew Howroyd, <a href="/A333580/b333580.txt">Table of n, a(n) for n = 1..378</a>
%H A333580 <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%F A333580 T(n,k) = T(k,n).
%e A333580 Square array T(n,k) begins:
%e A333580   1, 1,  1,   1,    1,     1,      1,      1, ...
%e A333580   1, 0,  1,   0,    1,     0,      1,      0, ...
%e A333580   1, 1,  2,   4,    8,    16,     32,     64, ...
%e A333580   1, 0,  4,   0,   20,     0,    111,      0, ...
%e A333580   1, 1,  8,  20,  104,   378,   1670,   6706, ...
%e A333580   1, 0, 16,   0,  378,     0,  10204,      0, ...
%e A333580   1, 1, 32, 111, 1670, 10204, 111712, 851073, ...
%e A333580   1, 0, 64,   0, 6706,     0, 851073,      0, ...
%o A333580 (Python)
%o A333580 # Using graphillion
%o A333580 from graphillion import GraphSet
%o A333580 import graphillion.tutorial as tl
%o A333580 def A333580(n, k):
%o A333580     if n == 1 or k == 1: return 1
%o A333580     universe = tl.grid(n - 1, k - 1)
%o A333580     GraphSet.set_universe(universe)
%o A333580     start, goal = 1, k * n
%o A333580     paths = GraphSet.paths(start, goal, is_hamilton=True)
%o A333580     return paths.len()
%o A333580 print([A333580(j + 1, i - j + 1) for i in range(12) for j in range(i + 1)])
%Y A333580 Rows n=1..10 (with 0 omitted) give: A000012, A000035, A011782(n-1), A014523, A014584, A333581, A333582, A333583, A333584, A333585.
%Y A333580 T(2*n-1,2*n-1) gives A001184(n-1).
%Y A333580 Cf. A271592.
%K A333580 nonn,tabl
%O A333580 1,13
%A A333580 _Seiichi Manyama_, Mar 27 2020