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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). 11
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
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.
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.
Main diagonal is A004003. Second and third rows give A001519, A188899.
Sequence in context: A176420 A099927 A139332 * A306344 A128612 A284731
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 May 26 03:59 EDT 2024. Contains 372807 sequences. (Running on oeis4.)