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
KEYWORD
tabf,uned,sign,less
AUTHOR
Roger L. Bagula and Gary W. Adamson, Jun 08 2008
STATUS
approved