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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). 10
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 (list; table; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois 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

Cf. A187596, A099390.

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

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Last modified March 4 19:52 EST 2021. Contains 341797 sequences. (Running on oeis4.)