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Search: id:A066170
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| A066170 |
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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. |
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+0
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; listen)
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OFFSET
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0,8
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COMMENT
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The table begins {1}; {-1, 1}; {1, -1, -1}; {-1, 2, 1, -1}; ...
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REFERENCES
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J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons, Croatica Chem. Acta, 77 (2004), 325-330. [See p. 328.]
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
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Polynomial recursion: p[k, x] = x*p[k - 1, x] - p[k - 2, x]. Also T(n, k) = (-1)^Floor[(k + 1)/2]*binomial[n - Floor[(k + 1)/2], Floor[k/2]] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
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EXAMPLE
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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; 2; 1; -1}.
Triangle (with rows reversed) begins:
1
1, -1
1, -1, -1,
1, -1, -2, 1
1, 1, -3, 2, 1
1, -1, -4, 3, 3,-1
1, -1, -5, 4, 6, -3, -1
1, -1, -6, 5, 10,-6, -4, 1
1, -1, -7, 6, 15, -10, -10, 4, 1
1, -1, -8, 7, 21, -15, -20, 10, 5,-1
1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1
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MATHEMATICA
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(* Triangular*) T[n_, k_] := (-1)^Floor[(k + 1)/2]*Binomial[n - Floor[(k + 1)/2], Floor[k/2]] Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
(* Polynomial recursion*) p[0, x] = 1; p[1, x] = x - 1; p[2, x] = x^2 - x - 1; p[3, x] = x^3 - x^2 - 2*x + 1; p[k_, x_] := p[k, x] = x*p[k - 1, x] - p[k - 2, x] ; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w] - Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 20 2006
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CROSSREFS
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Signed version of A046854.
Cf. A007700, A059455, A065941. For another version see A030111.
Adjacent sequences: A066167 A066168 A066169 this_sequence A066171 A066172 A066173
Sequence in context: A130461 A130777 A046854 this_sequence A071773 A000188 A097886
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Floor van Lamoen (fvlamoen(AT)hotmail.com), Dec 14 2001
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 02 2002
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