A187596 Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=0, n>=0).

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 3, 3, 1, 1, 1, 0, 5, 0, 5, 0, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 0, 13, 0, 36, 0, 13, 0, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 0, 34, 0, 281, 0, 281, 0, 34, 0, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 0, 89, 0, 2245, 0, 6728, 0, 2245, 0, 89, 0, 1, 1, 1, 144, 571, 6336

Offset: 0

Author

N. J. A. Sloane, Mar 11 2011

Keywords

• nonn
• tabl

Comments

A099390 supplemented by an initial row and column of 1's.

See A099390 (the main entry for this array) for further information.

If we work with the row index starting at 1 then every row of the array is a divisibility sequence, i.e., the terms satisfy the property that if n divides m then a(n) divide a(m) provided a(n) != 0. Row k satisfies a linear recurrence of order 2^floor(k/2) (Stanley, Ex. 36 p. 273). - Peter Bala, Apr 30 2014

Examples

			Array begins:
  1,  1,  1,  1,   1,    1,     1,     1,       1,      1,        1, ...
  1,  0,  1,  0,   1,    0,     1,     0,       1,      0,        1, ...
  1,  1,  2,  3,   5,    8,    13,    21,      34,     55,       89, ...
  1,  0,  3,  0,  11,    0,    41,     0,     153,      0,      571, ...
  1,  1,  5, 11,  36,   95,   281,   781,    2245,   6336,    18061, ...
  1,  0,  8,  0,  95,    0,  1183,     0,   14824,      0,   185921, ...
  1,  1, 13, 41, 281, 1183,  6728, 31529,  167089, 817991,  4213133, ...
  1,  0, 21,  0, 781,    0, 31529,     0, 1292697,      0, 53175517, ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

Links

• Alois P. Heinz, Antidiagonals n = 0..80, flattened
• James Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150 [math.CO], 1999.
• Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the second kind.
• Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Crossrefs

Cf. A099390.

See A187616 for a triangular version, and A187617, A187618 for the sub-array T(2m,2n).

See also A049310, A053117.

Programs

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
    elif mAlois P. Heinz, Apr 11 2011

Mathematica

t[m_, n_] := Product[2*(2+Cos[2*j*Pi/(m+1)]+Cos[2*k*Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}]; t[?OddQ, ?OddQ] = 0; Table[t[m-n, n] // FullSimplify, {m, 0, 13}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 07 2014, after A099390 *)

Formula

From Peter Bala, Apr 30 2014: (Start)

T(n,k)^2 = absolute value of Product_{b=1..k} Product_{a=1..n} ( 2*cos(a*Pi/(n+1)) + 2*i*cos(b*Pi/(k+1)) ), where i = sqrt(-1). See Propp, Section 5.

Equivalently, working with both the row index n and column index k starting at 1 we have T(n,k)^2 = absolute value of Resultant (F(n,x), U(k-1,x/2)), where U(n,x) is a Chebyshev polynomial of the second kind and F(n,x) is a Fibonacci polynomial defined recursively by F(0,x) = 0, F(1,x) = 1 and F(n,x) = x*F(n-1,x) + F(n-2,x) for n >= 2. The divisibility properties of the array entries mentioned in the Comments are a consequence of this result. (End)