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

1, 0, 2, 1, 0, 196, 0, 32, 0, 75272, 1, 0, 31329, 0, 599466256, 0, 450, 0, 135663392, 0, 28838245503008, 1, 0, 4941729, 0, 10956424382401, 0, 22463213552677201984, 0, 6272, 0, 233075146752, 0, 5652453608244879872, 0, 123818965842734619629420672

Offset: 0

Author

R. J. Mathar, Mar 21 2016

Keywords

• nonn
• tabl

Comments

Arrangements obtained by rotations and flips are counted as distinct.

Examples

			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.
Triangle begins:
1;
0, 2;
1, 0, 196;
0, 32, 0, 75272;
1, 0, 31329, 0, 599466256;
0, 450, 0, 135663392, 0, 28838245503008;
1, 0, 4941729, 0, 10956424382401, 0, 22463213552677201984;
		

Links

• R. J. Mathar, A270668: Domino tilings with one monomer in the center
• Index entries for sequences related to dominoes

Crossrefs

Cf. A098301, A143659 (diagonal), A189006 (free square in corner).

Formula

T(n,0) = A059841(n).

T(2n+1,1) = 2 * A098301(n+1). - Alois P. Heinz, Mar 21 2016

T(2n+1,1) = 2*A189006(2n+1,3)^2. - R. J. Mathar, Mar 22 2016

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

Extensions

More terms from Alois P. Heinz, Mar 21 2016