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%I #13 Dec 19 2022 10:33:16
%S -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,
%T -1,-1,-1,-56,-19,12,6,-4,-3,7,-13,10,-1,-1,1,119,26,-25,-3,12,5,-5,
%U -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
%N This sequence needs a meaningful name.
%C 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
%D Blackmore, D. and Kappraff, J. "Phyllotaxis and Toral Dynamical Systems." ZAMM (1995).
%D Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 75
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvesterMatrix.html">Sylvester Matrix</a>.
%e {-1, 0, -1, -1},
%e {1, 3, 4, 1, -1, -1},
%e {-1, -10, -8, 0, 3, 3, -1, -1},
%e {1, 25,13, -4, -5, -1, -2,6, -1, -1},
%e {-1, -56, -19, 12, 6, -4, -3, 7, -13, 10, -1, -1},
%e {1, 119, 26, -25, -3, 12, 5, -5, -18, 34, -32, 15, -1, -1},
%e {-1, -246, -34, 44, -8, -22, 0, 10, 7, 25, -81, 93, -61, 21, -1, -1},
%e {1, 501, 43, -70, 32, 30, -16, -18, 3, 5, -48, 166, -242, 200, -102, 28, -1, -1},
%e {-1, -1012, -53,104, -75, -28, 46, 20, -21, -20, 9,107, -348, 572, -574, 374, -157, 36, -1, -1},
%e {1, 2035, 64, -147,144, 3, -89, -2, 51, 19, -14, -29, -187, 735, -1314, 1502, -1177, 637, -228, 45, -1, -1}
%t 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 *)
%t p[x_, n_] := p[x.n] = x^n - Sum[x^i, {i, 0, n - 1}];
%t Table[SylvesterMatrix1[p[x, n], p[x, n - 1], x], {n, 2, 11}];
%t Table[Det[SylvesterMatrix1[p[x, n], p[x, n - 1], x]], {n, 2, 11}];
%t Table[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], {n, 2, 11}];
%t a = Table[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], x], {n, 2, 11}];
%t Flatten[a]
%K tabf,uned,sign,less
%O 2,6
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 08 2008
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