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Search: id:A139336
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| A139336 |
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Coefficient triangle of Sylvester resultant matrix characteristic polynomials of alternating signs: example matrix: {{-1, 1, -1, 1, 0}, {0, -1, 1, -1, 1}, {1, -1, 1, 0, 0}, {0, 1, -1, 1, 0}, {0,0, 1, -1, 1}}. |
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1
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| 1, -2, 3, -1, 1, 1, -2, -1, 1, -1, 1, -4, 10, -16, 19, -15, 7, -1, 1, 3, -1, -8, 1, 9, -4, -6, 1, -1, 1, -6, 21, -50, 90, -126, 141, -125, 85, -40, 11, -1, 1, 5, 4, -15, -19, 24, 29, -29, -20, 20, -6, -15, 1, -1, 1, -8, 36, -112, 266, -504, 784, -1016, 1107, -1015, 777, -483, 231, -77, 15, -1, 1, 7, 13, -14, -62, -2, 130, 28, -173
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums are: ( different from the cyclotomics)
{1, -1, 1, -5, 1, -21, 1, -85, 1, -341}:
Plotting the roots shows that they are all cyclotomic like of complex radius one:
w = Flatten[Table[Table[{Re[x], Im[x]} /. NSolve[p[x, n] == 0, x][[m]], {m, 1, n}], {n, 1, 10}], 1];
ListPlot[w]
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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.
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FORMULA
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p(x,n)=Sum((-1)^i*x^i,{i,0,n)); M(n)=SylvesterMatrix(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, 1, -2, -1, 1, -1},
{1, -4,10, -16, 19, -15, 7, -1},
{1, 3, -1, -8, 1, 9, -4, -6, 1, -1},
{1, -6, 21, -50, 90, -126,141, -125, 85, -40, 11, -1},
{1, 5, 4, -15, -19, 24, 29, -29, -20, 20, -6, -15, 1, -1},
{1, -8, 36, -112,266, -504, 784, -1016, 1107, -1015, 777, -483, 231, -77, 15, -1},
{1, 7, 13, -14, -62, -2, 130, 28, -173, -19, 154, -28, -98, 14, -8, -28, 1, -1},
{1, -10, 55, -210, 615, -1452, 2850, -4740, 6765, -8350, 8953, -8349, 6756, -4704,2766, -1326, 489, -126, 19, -1},
{1, 9, 26,3, -116, -131, 229, 400, -305, -655, 364, 681, -417, -441, 342, 54, -285, -45, -10, -45, 1, -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[(-1)^i*x^i, {i, 0, n}]; Table[SylvesterMatrix1[p[x, n], p[x, n - 1], x], {n, 2, 11}]; Table[Det[SylvesterMatrix1[p[x, n], p[x, n - 1], x]], {n, 2, 11}]; Table[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], {n, 2, 11}] a = Table[C; oefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[ x, n], p[x, n - 1], x], x], x], {n, 2, 11}]; Flatten[a]
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CROSSREFS
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Sequence in context: A122587 A086195 A086197 this_sequence A100619 A094006 A140188
Adjacent sequences: A139333 A139334 A139335 this_sequence A139337 A139338 A139339
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KEYWORD
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uned,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 09 2008
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