A250662 Number A(n,k) of tilings of a 2k X n rectangle using 2n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 6, 36, 1, 1, 1, 1, 1, 1, 13, 95, 1, 1, 1, 1, 1, 1, 7, 22, 281, 1, 1, 1, 1, 1, 1, 1, 15, 64, 781, 1, 1, 1, 1, 1, 1, 1, 8, 25, 155, 2245, 1, 1, 1, 1, 1, 1, 1, 1, 17, 37, 321, 6336, 1, 1

Offset: 0

Author

Alois P. Heinz, Nov 26 2014

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1, 1,    1,   1,   1,  1,  1,  1,  1, ...
  1, 1,    1,   1,   1,  1,  1,  1,  1, ...
  1, 1,    5,   1,   1,  1,  1,  1,  1, ...
  1, 1,   11,   6,   1,  1,  1,  1,  1, ...
  1, 1,   36,  13,   7,  1,  1,  1,  1, ...
  1, 1,   95,  22,  15,  8,  1,  1,  1, ...
  1, 1,  281,  64,  25, 17,  9,  1,  1, ...
  1, 1,  781, 155,  37, 28, 19, 10,  1, ...
  1, 1, 2245, 321, 100, 41, 31, 21, 11, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..100, flattened
• Wikipedia, Polyomino

Crossrefs

Columns k=0+1,2-10 give: A000012, A005178(n+1), A236577, A236582, A247117, A250663, A250664, A250665, A250666, A250667.

Cf. A251072.

Programs

Maple

b:= proc(n, l) option remember; local d, k; d:= nops(l)/2;
      if n=0 then 1
    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
    else for k while l[k]>0 do od;
         `if`(nd+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][],1$d,l[k+d..2*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$2*k])):
seq(seq(A(n,d-n), n=0..d), d=0..14);

Mathematica

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, k}, Which[n == 0, 1, Min[l] > 0 , Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0, k++]; If[n d]]] + If[d == 1 || k > d+1 || Max[l[[k ;; k+d-1]]] > 0, 0, b[n, Join[l[[1 ;; k-1]], Array[1&, d], l[[k+d ;; 2*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 2k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)