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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). 13
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
James Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150 [math.CO], 1999.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the second kind.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
FORMULA
From Peter Bala, Apr 30 2014: (Start)
T(n,k)^2 = absolute value of Product_{b=1..k} Product_{a=1..n} ( 2*cos(a*Pi/(n+1)) + 2*i*cos(b*Pi/(k+1)), 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-François 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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 11 2011
STATUS
approved

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Last modified May 20 06:19 EDT 2024. Contains 372703 sequences. (Running on oeis4.)