Chance of Hamiltonian Path in Sudoku cell

I just wrote a program to test the following $m \times n$ boards. The coordinates are (paths from 1 to $mn$, cycles).

\begin{array}{c|cccc} m \backslash n & 1 & 2 & 3 & 4 & 5 \\ \hline \\ 1 & (1,1) & (2,2) & (2,0) & (2,0) & (2, 0) \\ 2 & (2,2) & (24,24) & (96,48) & (416,128) & (1536,320) \\ 3 & (2,0) & (96,48) & (784,288) & \\ 4 & (2,0) & (416,128) & & \\ 5 & (2,0) & (1536,320)\\ \end{array}

These are OEIS sequences A158651 for paths and A140519 for cycles (where the latter needs to be multiplied by $2N^2$ since they only count the cycles and not their realisations through permutations of numbers). The entries refer to the paper Enumerating Hamiltonian Cycles by Ville H. Pettersson in the Electronic Journal of Combinatorics, Volume $21$, Issue $4$, $2014$.