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A270668 Triangle read by rows: The number of domino tilings of the (2n+1) X (2m+1) board with a central free square.

%I A270668 #23 Apr 13 2016 17:25:00
%S A270668 1,0,2,1,0,196,0,32,0,75272,1,0,31329,0,599466256,0,450,0,135663392,0,
%T A270668 28838245503008,1,0,4941729,0,10956424382401,0,22463213552677201984,0,
%U A270668 6272,0,233075146752,0,5652453608244879872,0,123818965842734619629420672
%N A270668 Triangle read by rows: The number of domino tilings of the (2n+1) X (2m+1) board with a central free square.
%C A270668 Arrangements obtained by rotations and flips are counted as distinct.
%H A270668 R. J. Mathar, <a href="/A270668/a270668.pdf">A270668: Domino tilings with one monomer in the center</a>
%H A270668 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A270668 T(n,0) = A059841(n).
%F A270668 T(2n+1,1) = 2 * A098301(n+1). - _Alois P. Heinz_, Mar 21 2016
%F A270668 T(2n+1,1) = 2*A189006(2n+1,3)^2. - _R. J. Mathar_, Mar 22 2016
%F A270668 Conjectured g.f. for column 3: ( -1 -4*x +543*x^2 -6238*x^3 +17032*x^4 -6238*x^5 +543*x^6 -4*x^7 -x^8 ) / ( (x-1) *(x^2-7*x+1) *(x^2-23*x+1) *(x^4 -161*x^3 +576*x^2 -161*x +1) ). - _R. J. Mathar_, Mar 23 2016
%e A270668 For n=m=1, the 3 X 3 board can be covered in T(1,1)=2 ways, starting in one corner with either a horizontal or a vertical domino.
%e A270668 Triangle begins:
%e A270668 1;
%e A270668 0, 2;
%e A270668 1, 0, 196;
%e A270668 0, 32, 0, 75272;
%e A270668 1, 0, 31329, 0, 599466256;
%e A270668 0, 450, 0, 135663392, 0, 28838245503008;
%e A270668 1, 0, 4941729, 0, 10956424382401, 0, 22463213552677201984;
%Y A270668 Cf. A098301, A143659 (diagonal), A189006 (free square in corner).
%K A270668 nonn,tabl
%O A270668 0,3
%A A270668 _R. J. Mathar_, Mar 21 2016
%E A270668 More terms from _Alois P. Heinz_, Mar 21 2016