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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).

%I A187616 #28 Mar 14 2015 22:58:53
%S A187616 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,
%T A187616 1,0,21,0,781,0,31529,0,1,1,34,153,2245,14824,167089,1292697,12988816,
%U A187616 1,0,55,0,6336,0,817991,0,108435745,0,1,1,89,571,18061,185921,4213133,53175517,1031151241,14479521761,258584046368
%N A187616 Triangle T(m,n) read by rows: number of domino tilings of the m X n grid (0 <= m <= n).
%C A187616 A099390 is the main entry for this problem.
%C A187616 Triangle read by rows: the square array in A187596 with entries above main diagonal deleted.
%H A187616 Alois P. Heinz, <a href="/A187616/b187616.txt">Rows n = 1..32, flattened</a>
%H A187616 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%e A187616 Triangle begins:
%e A187616 1
%e A187616 1 0
%e A187616 1 1  2
%e A187616 1 0  3   0
%e A187616 1 1  5  11   36
%e A187616 1 0  8   0   95     0
%e A187616 1 1 13  41  281  1183   6728
%e A187616 1 0 21   0  781     0  31529       0
%e A187616 1 1 34 153 2245 14824 167089 1292697 12988816
%e A187616 ...
%p A187616 with(LinearAlgebra):
%p A187616 T:= proc(m,n) option remember; local i, j, t, M;
%p A187616       if m<=1 or n<=1 then 1 -irem(n*m, 2)
%p A187616     elif irem(n*m, 2)=1 then 0
%p A187616     else M:= Matrix(n*m, shape =skewsymmetric);
%p A187616          for i to n do
%p A187616            for j to m do
%p A187616              t:= (i-1)*m+j;
%p A187616              if j<m then M[t, t+1]:= 1 fi;
%p A187616              if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi
%p A187616            od
%p A187616          od;
%p A187616          sqrt(Determinant(M))
%p A187616       fi
%p A187616     end:
%p A187616 seq(seq(T(m, n), n=0..m), m=0..10);  # _Alois P. Heinz_, Apr 11 2011
%t A187616 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 *)
%Y A187616 Cf. A099390, A187596. See A099390 for sequences appearing in the rows and columns. See also A187617, A187618.
%K A187616 nonn,tabl
%O A187616 0,6
%A A187616 _N. J. A. Sloane_, Mar 11 2011