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A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

%I A187617 #48 Oct 22 2021 23:47:25
%S A187617 1,1,1,1,2,1,1,5,5,1,1,13,36,13,1,1,34,281,281,34,1,1,89,2245,6728,
%T A187617 2245,89,1,1,233,18061,167089,167089,18061,233,1,1,610,145601,4213133,
%U A187617 12988816,4213133,145601,610,1
%N A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).
%C A187617 A099390 is the main entry for this problem.
%C A187617 The even-indexed rows and columns of the square array in A187596.
%C A187617 Row (and column) 2 is given by A122367. - _Nathaniel Johnston_, Mar 22 2011
%H A187617 Alois P. Heinz, <a href="/A187617/b187617.txt">Antidiagonals n = 0..26, flattened</a>
%H A187617 N. Allegra, <a href="http://arxiv.org/abs/1410.4131">Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics</a>, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.
%H A187617 Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, <a href="https://doi.org/10.37236/4472">Sandpiles and Dominos</a>, El. J. Comb., 22 (2015), P1.66. See Theorem 15.
%H A187617 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%e A187617 The array begins:
%e A187617   1,  1,     1,       1,          1,            1, ...
%e A187617   1,  2,     5,      13,         34,           89, ...
%e A187617   1,  5,    36,     281,       2245,        18061, ...
%e A187617   1, 13,   281,    6728,     167089,      4213133, ...
%e A187617   1, 34,  2245,  167089,   12988816,   1031151241, ...
%e A187617   1, 89, 18061, 4213133, 1031151241, 258584046368, ...
%p A187617 ft:=(m,n)->
%p A187617 2^(m*n/2)*mul( mul(
%p A187617 (cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
%p A187617 T:=(m,n)->round(evalf(ft(m,n),300));
%t A187617 T[m_, n_] := Product[2(2 + Cos[(2j Pi)/(2m+1)] + Cos[(2k Pi)/(2n+1)]), {j, 1, m}, {k, 1, n}];
%t A187617 Table[T[m-n, n] // Round, {m, 0, 8}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Aug 05 2018 *)
%o A187617 (PARI) default(realprecision, 120);
%o A187617 {T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*cos(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))} \\ _Seiichi Manyama_, Jan 09 2021
%Y A187617 A187618 is the triangle version.
%Y A187617 Cf. A187596, A099390, A348566.
%Y A187617 Main diagonal is A004003. Second and third rows give A001519, A188899.
%K A187617 nonn,tabl
%O A187617 0,5
%A A187617 _N. J. A. Sloane_, Mar 11 2011
%E A187617 More terms from _Nathaniel Johnston_, Mar 22 2011