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

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