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A139344
This sequence needs a meaningful name.
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
OFFSET
2,6
COMMENTS
Row n appears to be the expansion of the characteristic polynomial of the Sylvester matrix of (p(x, n), p(x, n - 1)) where p(x, n) = x^n - Sum_{i=0..n-1} x^i. Offset may actually be 2. - Joerg Arndt, Dec 19 2022
REFERENCES
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
Eric Weisstein's World of Mathematics, Sylvester Matrix.
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
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]] ] (* from https://mathworld.wolfram.com/SylvesterMatrix.html *)
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
KEYWORD
tabf,uned,sign,less
AUTHOR
STATUS
approved