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A187616 Triangle T(m,n) read by rows: number of domino tilings of the m X n grid (0 <= m <= n). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A165252 A127373 A200123 * A217262 A260616 A284950

Adjacent sequences:  A187613 A187614 A187615 * A187617 A187618 A187619

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Mar 11 2011

STATUS

approved

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Last modified February 20 20:52 EST 2019. Contains 320345 sequences. (Running on oeis4.)