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A066170
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Triangle read by rows: T(n,k) = (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n, n >= 0.
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29
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1, -1, 1, 1, -1, -1, -1, 2, 1, -1, 1, -2, -3, 1, 1, -1, 3, 3, -4, -1, 1, 1, -3, -6, 4, 5, -1, -1, -1, 4, 6, -10, -5, 6, 1, -1, 1, -4, -10, 10, 15, -6, -7, 1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, 1, -5, -15, 20, 35, -21, -28, 8, 9, -1, -1, -1, 6, 15, -35, -35, 56, 28, -36, -9, 10, 1, -1, 1, -6, -21, 35, 70, -56, -84, 36, 45, -10, -11
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OFFSET
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0,8
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COMMENTS
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The original name of this sequence was: Triangle giving coefficients of characteristic function of n X n matrix in which the left upper half and the antidiagonal are filled with 1's and the right lower half is filled with 0's. As was pointed out by L. Edson Jeffery this is only correct if we multiply each triangle row by (-1)^n. For the straightforward version of the coefficients of the characteristic polynomials see A187660. - Johannes W. Meijer, Aug 08 2011
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REFERENCES
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Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
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LINKS
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FORMULA
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T(n,k) = (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k);
T(n,k) = (-1)^n*A187660(n,k). (End)
abs(T(n,n-k)) = A065941(n,k). (End)
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EXAMPLE
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The table begins {1}; {-1, 1}; {1, -1, -1}; {-1, 2, 1, -1}; ...
The characteristic function of
( 1 1 1 )
( 1 1 0 )
( 1 0 0 )
is f(x) = x^3 - 2x^2 - x + 1, so the 3rd row is (-1)^3 times the f(x) coefficients, i.e., {-1; 2; 1; -1}.
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MAPLE
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MATHEMATICA
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Flatten[Table[(-1)^n*(-1)^Floor[3*k/2]*Binomial[Floor[(n+k)/2], k], {n, 0, 12}, {k, 0, n}]] (* Indranil Ghosh, Feb 19 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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