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

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, 2245, 89, 1, 1, 233, 18061, 167089, 167089, 18061, 233, 1, 1, 610, 145601, 4213133, 12988816, 4213133, 145601, 610, 1

Offset

0,5

Comments

A099390 is the main entry for this problem.

The even-indexed rows and columns of the square array in A187596.

Row (and column) 2 is given by A122367. - Nathaniel Johnston, Mar 22 2011

Links

Alois P. Heinz, Antidiagonals n = 0..26, flattened

N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 15.

Index entries for sequences related to dominoes

Example

The array begins:
  1,  1,     1,       1,          1,            1, ...
  1,  2,     5,      13,         34,           89, ...
  1,  5,    36,     281,       2245,        18061, ...
  1, 13,   281,    6728,     167089,      4213133, ...
  1, 34,  2245,  167089,   12988816,   1031151241, ...
  1, 89, 18061, 4213133, 1031151241, 258584046368, ...

Maple

ft:=(m, n)->
2^(m*n/2)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
T:=(m, n)->round(evalf(ft(m, n), 300));

Mathematica

T[m_, n_] := Product[2(2 + Cos[(2j Pi)/(2m+1)] + Cos[(2k Pi)/(2n+1)]), {j, 1, m}, {k, 1, n}];
Table[T[m-n, n] // Round, {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Aug 05 2018 *)

Prog

(PARI) default(realprecision, 120);
{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

Crossrefs

A187618 is the triangle version.

Cf. A187596, A099390, A348566.

Main diagonal is A004003. Second and third rows give A001519, A188899.

Sequence in context: A176420 A099927 A139332 * A306344 A128612 A284731

Adjacent sequences: A187614 A187615 A187616 * A187618 A187619 A187620

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 11 2011

Extensions

More terms from Nathaniel Johnston, Mar 22 2011

Status

approved