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A139344 A triangular sequence of polynomials of the coefficients of the characteristic polynomials of the Sylvester resultant matrices of the Bonacci polynomials: example matrix:for x^2-x-1 and x-1; {{1, -1, -1}, {1, -1, 0}, {0, 1, -1}}. 0
-1, 0, -1, -1, 1, 3, 4, 1, -1, -1, -1, -10, -8, 0, 3, 3, -1, -1, 1, 25, 13, -4, -5, -1, -2, 6, -1, -1, -1, -56, -19, 12, 6, -4, -3, 7, -13, 10, -1, -1, 1, 119, 26, -25, -3, 12, 5, -5, -18, 34, -32, 15, -1, -1, -1, -246, -34, 44, -8, -22, 0, 10, 7, 25, -81, 93, -61, 21, -1, -1, 1, 501, 43, -70, 32, 30, -16, -18, 3, 5, -48, 166, -242, 200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Row sums are:

{-3, 7, -15, 31, -63, 127, -255, 511, -1023, 2047}

If the polynomials have a common factor the determinant of the matrix is zero.

I use the matrix making software from the MathWorld page.

REFERENCES

Weisstein, Eric W. "Sylvester Matrix." http://mathworld.wolfram.com/SylvesterMatrix.html

Blackmore, D. and Kappraff, J. "Phyllotaxis and Toral Dynamical Systems." ZAMM (1995).

Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 75

LINKS

Table of n, a(n) for n=1..84.

FORMULA

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))).

EXAMPLE

{-1, 0, -1, -1},

{1, 3, 4, 1, -1, -1},

{-1, -10, -8, 0, 3, 3, -1, -1},

{1, 25,13, -4, -5, -1, -2,6, -1, -1},

{-1, -56, -19, 12, 6, -4, -3, 7, -13, 10, -1, -1},

{1, 119, 26, -25, -3, 12, 5, -5, -18, 34, -32, 15, -1, -1},

{-1, -246, -34, 44, -8, -22, 0, 10, 7, 25, -81, 93, -61, 21, -1, -1},

{1, 501, 43, -70, 32, 30, -16, -18, 3, 5, -48, 166, -242, 200, -102, 28, -1, -1},

{-1, -1012, -53,104, -75, -28, 46, 20, -21, -20, 9,107, -348, 572, -574, 374, -157, 36, -1, -1},

{1, 2035, 64, -147,144, 3, -89, -2, 51, 19, -14, -29, -187, 735, -1314, 1502, -1177, 637, -228, 45, -1, -1}

MATHEMATICA

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] = x^n - Sum[x^i, {i, 0, n - 1}]; 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[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], x], {n, 2, 11}]; Flatten[a]

CROSSREFS

Sequence in context: A299989 A058022 A215202 * A137925 A171528 A299924

Adjacent sequences:  A139341 A139342 A139343 * A139345 A139346 A139347

KEYWORD

tabf,uned,sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Jun 08 2008

STATUS

approved

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Last modified November 13 17:34 EST 2019. Contains 329106 sequences. (Running on oeis4.)