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A187596 Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=0, n>=0). 12
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

A099390 supplemented by an initial row and column of 1's.

See A099390 (the main entry for this array) for further information.

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

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

LINKS

Alois P. Heinz, Antidiagonals n = 0..80, flattened

J. Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150v2 [math.CO]

E. W. Weisstein, Chebyshev Polynomial of the second kind MathWorld

E. W. Weisstein, Fibonacci Polynomial MathWorld

FORMULA

From Peter Bala, Apr 30 2014: (Start)

T(n,k)^2 = absolute value of Prod(Prod( 2*cos(a*Pi/(n+1)) + 2*i*cos(b*Pi/(k+1)), a = 1..n), b = 1..k), where i = sqrt(-1). See Propp, Section 5.

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)

EXAMPLE

Array begins:

  1,  1,  1,  1,   1,    1,     1,     1,       1,      1,        1, ...

  1,  0,  1,  0,   1,    0,     1,     0,       1,      0,        1, ...

  1,  1,  2,  3,   5,    8,    13,    21,      34,     55,       89, ...

  1,  0,  3,  0,  11,    0,    41,     0,     153,      0,      571, ...

  1,  1,  5, 11,  36,   95,   281,   781,    2245,   6336,    18061, ...

  1,  0,  8,  0,  95,    0,  1183,     0,   14824,      0,   185921, ...

  1,  1, 13, 41, 281, 1183,  6728, 31529,  167089, 817991,  4213133, ...

  1,  0, 21,  0, 781,    0, 31529,     0, 1292697,      0, 53175517, ...

MAPLE

with(LinearAlgebra):

T:= proc(m, n) option remember; local i, j, t, M;

      if m<=1 or n<=1 then 1 -irem(n*m, 2)

    elif irem(n*m, 2)=1 then 0

    elif m<n then T(n, m)

    else M:= Matrix(n*m, shape=skewsymmetric);

         for i to n do

           for j to m do

             t:= (i-1)*m+j;

             if j<m then M[t, t+1]:= 1 fi;

             if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi

           od

         od;

         sqrt(Determinant(M))

      fi

    end:

seq(seq(T(m, d-m), m=0..d), d=0..14);  # Alois P. Heinz, Apr 11 2011

MATHEMATICA

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-Fran├žois Alcover, Jan 07 2014, after A099390 *)

CROSSREFS

Cf. A099390.

See A187616 for a triangular version, and A187617, A187618 for the sub-array T(2m,2n).

See also A049310, A053117.

Sequence in context: A035467 A254045 A024996 * A263863 A134655 A262124

Adjacent sequences:  A187593 A187594 A187595 * A187597 A187598 A187599

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Mar 11 2011

STATUS

approved

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Last modified March 4 20:58 EST 2021. Contains 341811 sequences. (Running on oeis4.)