A187616 Triangle T(m,n) read by rows: number of domino tilings of the m X n grid (0 <= m <= n).

1, 1, 0, 1, 1, 2, 1, 0, 3, 0, 1, 1, 5, 11, 36, 1, 0, 8, 0, 95, 0, 1, 1, 13, 41, 281, 1183, 6728, 1, 0, 21, 0, 781, 0, 31529, 0, 1, 1, 34, 153, 2245, 14824, 167089, 1292697, 12988816, 1, 0, 55, 0, 6336, 0, 817991, 0, 108435745, 0, 1, 1, 89, 571, 18061, 185921, 4213133, 53175517, 1031151241, 14479521761, 258584046368

Offset

0,6

Comments

A099390 is the main entry for this problem.

Triangle read by rows: the square array in A187596 with entries above main diagonal deleted.

Links

Alois P. Heinz, Rows n = 1..32, flattened

Index entries for sequences related to dominoes

Example

Triangle begins:
1
1 0
1 1  2
1 0  3   0
1 1  5  11   36
1 0  8   0   95     0
1 1 13  41  281  1183   6728
1 0 21   0  781     0  31529       0
1 1 34 153 2245 14824 167089 1292697 12988816
...

Maple

with(LinearAlgebra):
T:= proc(m, n) option remember; local i, j, t, M;
      if m<=1 or n<=1 then 1 -irem(n*m, 2)
    elif irem(n*m, 2)=1 then 0
    else M:= Matrix(n*m, shape =skewsymmetric);
         for i to n do
           for j to m do
             t:= (i-1)*m+j;
             if j<m then M[t, t+1]:= 1 fi;
             if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi
           od
         od;
         sqrt(Determinant(M))
      fi
    end:
seq(seq(T(m, n), n=0..m), m=0..10);  # Alois P. Heinz, Apr 11 2011

Mathematica

T[m_, n_] := T[m, n] = Module[{i, j, t, M}, Which[m <= 1 || n <= 1, 1 - Mod[n*m, 2], Mod[n*m, 2] == 1, 0, True, M[i_, j_] /; j < i := -M[j, i]; M[_, _] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j; If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1 - 2*Mod[j, 2]]]]; Sqrt[Det[Table[M[i, j], {i, 1, n*m}, {j, 1, n*m}]]]]]; Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}] // Flatten (* Jean-François Alcover, Jan 07 2014, translated from Maple *)

Crossrefs

Cf. A099390, A187596. See A099390 for sequences appearing in the rows and columns. See also A187617, A187618.

Sequence in context: A371954 A127373 A200123 * A217262 A378148 A260616

Adjacent sequences: A187613 A187614 A187615 * A187617 A187618 A187619

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 11 2011

Status

approved