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A189002 Number of domino tilings of the n X n grid with upper left corner removed iff n is odd.

%I A189002 #21 Jul 14 2020 15:26:35
%S A189002 1,1,2,4,36,192,6728,100352,12988816,557568000,258584046368,
%T A189002 32565539635200,53060477521960000,19872369301840986112,
%U A189002 112202208776036178000000,126231322912498539682594816,2444888770250892795802079170816,8326627661691818545121844900397056
%N A189002 Number of domino tilings of the n X n grid with upper left corner removed iff n is odd.
%H A189002 Alois P. Heinz, <a href="/A189002/b189002.txt">Table of n, a(n) for n = 0..50</a>
%H A189002 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A189002 a(n) = A189006(n,n).
%e A189002 a(3) = 4 because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
%e A189002   . .___.  . .___.  . .___.  . .___.
%e A189002   ._|___|  ._|___|  ._| | |  ._|___|
%e A189002   | |___|  | | | |  | |_|_|  |___| |
%e A189002   |_|___|  |_|_|_|  |_|___|  |___|_|
%t A189002 A[1, 1] = 1;
%t A189002 A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; 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 - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
%t A189002 a[n_] := A[n, n];
%t A189002 a /@ Range[0, 17] (* _Jean-François Alcover_, Feb 27 2020, after _Alois P. Heinz_ in A189006 *)
%Y A189002 Main diagonal of A189006.
%Y A189002 Bisection gives: A004003 (even part), A007341 (odd part).
%K A189002 nonn
%O A189002 0,3
%A A189002 _Alois P. Heinz_, Apr 15 2011