(* Vaclav Kotesovec 2011, based on article by R. Tauraso, 2006 *)
<< Combinatorica`
(* semi=2 Rook + Leaper[d,d] *)
(* semi=1 Rook + semi-Leaper[d,d] *)
(* version for d=2 *)
semi = 1; d = 2; Table[If[n == 1, 1, suma = 0;
   n1 = 0; Do[If[Mod[i - 1, d] == 0, n1 = n1 + 1], {i, 1, n}];
   n2 = 0; Do[If[Mod[i - 2, d] == 0, n2 = n2 + 1], {i, 1, n}];
   Do[Do[Do[
        Do[part1 = (-1)^(r1 + r2)*
            semi^(c1 + c2)*(n - (r1 + r2) - (c1 + c2))!*
            Binomial[n1 - r1, c1]*Binomial[n2 - r2, c2];
          
          If[c1 == 0, If[r1 == 0, nl1 = 1, nl1 = 0];, 
           nl1 = NumberOfCompositions[r1 - c1, c1];];
          
          If[c2 == 0, If[r2 == 0, nl2 = 1, nl2 = 0];, 
           nl2 = NumberOfCompositions[r2 - c2, c2];];
          Do[Do[
             
             If[c1 == 0, z1 = {};, 
              z1 = Compositions[r1 - c1, c1][[l1]];];
             
             If[c2 == 0, z2 = {};, 
              z2 = Compositions[r2 - c2, c2][[l2]];];
             z = Flatten[{z1, z2}];
             
             Do[m = NthSubset[p, Range[c1 + c2]]; u = Length[m]; 
              If[u == 0, s = u, s = u + Sum[z[[m[[i]]]], {i, 1, u}];];
              
              If[n1 - s >= 0 && n2 - (r1 + r2 - s) >= 0, 
               suma = suma + 
                  part1*Binomial[n1 - s, u]*u!*
                   Binomial[n2 - (r1 + r2 - s), 
                    c1 + c2 - u]*(c1 + c2 - u)!;];
              , {p, 0, 2^(c1 + c2) - 1}];
             , {l2, 1, nl2}];
           , {l1, 1, nl1}];
          , {c2, 0, Min[r2, n2 - r2]}];
        , {c1, 0, Min[r1, n1 - r1]}];
      , {r2, 0, n2 - 1}];
    , {r1, 0, n1 - 1}];
   suma]
  , {n, 1, 18}]