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A139343
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A triangular sequence of polynomials of the coefficients of the characteristic polynomials of the Sylvester resultant matrices of the Cyclotomic polynomials: example matrix:for x^2+x+1 and x+1; {{1, 1, 1}, {1, 1, 0}, {0, 1, 1}}.
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0
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1, -2, 3, -1, 1, -3, 6, -7, 5, -1, 1, -4, 10, -16, 19, -15, 7, -1, 1, -5, 15, -30, 45, -51, 44, -26, 9, -1, 1, -6, 21, -50, 90, -126, 141, -125, 85, -40, 11, -1, 1, -7, 28, -77, 161, -266, 357, -393, 356, -260, 146, -57, 13, -1, 1, -8, 36, -112, 266, -504, 784, -1016, 1107, -1015, 777, -483, 231, -77, 15, -1, 1, -9, 45, -156
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OFFSET
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1,2
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COMMENTS
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Row sums are one and so are the determinants.
These matrices start at 3x3 and jump by 2 at each step.
If the polynomials have a common factor, the determinant of the matrix is zero.
I use the matrix making software from the MathWorld page.
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REFERENCES
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Weisstein, Eric W. "Sylvester Matrix." http://mathworld.wolfram.com/SylvesterMatrix.html
Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 232
Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 75
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LINKS
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Table of n, a(n) for n=1..74.
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FORMULA
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p(x,n)=Sum(x^i,{i,0,n-1); M(n)=SylvesterMatix( p(x,n),p(x,n-1); out_n,m=Coefficients(Characteristicpolynomial(M(n))).
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EXAMPLE
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{1, -2, 3, -1},
{1, -3, 6, -7, 5, -1},
{1, -4, 10, -16, 19, -15, 7, -1},
{1, -5, 15, -30, 45, -51, 44, -26, 9, -1},
{1, -6, 21, -50, 90, -126, 141, -125, 85, -40, 11, -1},
{1, -7, 28, -77,161, -266, 357, -393, 356, -260, 146, -57, 13, -1},
{1, -8, 36, -112, 266, -504, 784, -1016, 1107, -1015,777, -483, 231, -77, 15, -1},
{1, -9, 45, -156, 414, -882, 1554, -2304,2907, -3139, 2906, -2296, 1526, -826, 344, -100, 17, -1},
{1, -10, 55, -210, 615, -1452, 2850, -4740, 6765, -8350, 8953, -8349, 6756, -4704, 2766, -1326, 489, -126, 19, -1}
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MATHEMATICA
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Clear[p, x] SylvesterMatrix1[poly1_, poly2_, var_] := Function[{coeffs1, coeffs2}, With[ {l1 = Length[coeffs1], l2 = Length[coeffs2]}, Join[ NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2], NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2] ] ] ][ Reverse[CoefficientList[poly1, var]], Reverse[CoefficientList[poly2, var]] ] p[x_, n_] := p[x.n] = Sum[x^i, {i, 0, n - 1}]; Table[Det[SylvesterMatrix1[p[x, n], p[x, n - 1], x]], {n, 3, 11}]; Table[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[ x, n - 1], x], x], {n, 3, 11}]; a = Table[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], x], {n, 3, 11}]; Flatten[a]
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CROSSREFS
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Sequence in context: A139438 A135392 A071947 * A059247 A194518 A023572
Adjacent sequences: A139340 A139341 A139342 * A139344 A139345 A139346
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KEYWORD
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uned,tabf,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Jun 08 2008
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STATUS
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approved
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