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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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28
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| 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
| Henry W. Gould, "A Variant of Pascal's Triangle", The Fibonacci Quarterly,3;4 Dec. 1965, pp. 257-271.
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
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FORMULA
| From L. Edson Jeffery, Mar 23 2011: (Start)
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)
From Johannes W. Meijer, Aug 08 2011: (Start)
abs(T(n,k)) = A046854(n,k) = abs(A108299(n,n-k))
abs(T(n,n-k)) = A065941(n,k) (End)
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EXAMPLE
| 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
| A066170 := proc(n, k): (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2), k) end: seq(seq(A066170(n, k), k=0..n), n=0..11); [Johannes W. Meijer, Aug 08 2011]
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CROSSREFS
| Cf. A007700, A059455, A065941. For another version see A030111.
Sequence in context: A130777 A046854 A187660 * A184957 A205792 A071773
Adjacent sequences: A066167 A066168 A066169 * A066171 A066172 A066173
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Floor van Lamoen (fvlamoen(AT)hotmail.com), Dec 14 2001
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 02 2002
Corrected and edited by Johannes W. Meijer, Aug 08 2011
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