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%I A187596 #63 Aug 06 2022 07:23:59
%S A187596 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,
%T A187596 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,
%U A187596 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
%N A187596 Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=0, n>=0).
%C A187596 A099390 supplemented by an initial row and column of 1's.
%C A187596 See A099390 (the main entry for this array) for further information.
%C A187596 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
%D A187596 R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.
%H A187596 Alois P. Heinz, <a href="/A187596/b187596.txt">Antidiagonals n = 0..80, flattened</a>
%H A187596 James Propp, <a href="http://arxiv.org/abs/math/9904150">Enumeration of Matchings: Problems and Progress</a>, arXiv:math/9904150 [math.CO], 1999.
%H A187596 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">Chebyshev Polynomial of the second kind</a>.
%H A187596 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.
%F A187596 From _Peter Bala_, Apr 30 2014: (Start)
%F A187596 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.
%F A187596 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)
%e A187596 Array begins:
%e A187596   1,  1,  1,  1,   1,    1,     1,     1,       1,      1,        1, ...
%e A187596   1,  0,  1,  0,   1,    0,     1,     0,       1,      0,        1, ...
%e A187596   1,  1,  2,  3,   5,    8,    13,    21,      34,     55,       89, ...
%e A187596   1,  0,  3,  0,  11,    0,    41,     0,     153,      0,      571, ...
%e A187596   1,  1,  5, 11,  36,   95,   281,   781,    2245,   6336,    18061, ...
%e A187596   1,  0,  8,  0,  95,    0,  1183,     0,   14824,      0,   185921, ...
%e A187596   1,  1, 13, 41, 281, 1183,  6728, 31529,  167089, 817991,  4213133, ...
%e A187596   1,  0, 21,  0, 781,    0, 31529,     0, 1292697,      0, 53175517, ...
%p A187596 with(LinearAlgebra):
%p A187596 T:= proc(m,n) option remember; local i, j, t, M;
%p A187596       if m<=1 or n<=1 then 1 -irem(n*m, 2)
%p A187596     elif irem(n*m, 2)=1 then 0
%p A187596     elif m<n then T(n,m)
%p A187596     else M:= Matrix(n*m, shape=skewsymmetric);
%p A187596          for i to n do
%p A187596            for j to m do
%p A187596              t:= (i-1)*m+j;
%p A187596              if j<m then M[t, t+1]:= 1 fi;
%p A187596              if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi
%p A187596            od
%p A187596          od;
%p A187596          sqrt(Determinant(M))
%p A187596       fi
%p A187596     end:
%p A187596 seq(seq(T(m, d-m), m=0..d), d=0..14);  # _Alois P. Heinz_, Apr 11 2011
%t A187596 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 *)
%Y A187596 Cf. A099390.
%Y A187596 See A187616 for a triangular version, and A187617, A187618 for the sub-array T(2m,2n).
%Y A187596 See also A049310, A053117.
%K A187596 nonn,tabl
%O A187596 0,13
%A A187596 _N. J. A. Sloane_, Mar 11 2011

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