%I A139336
%S A139336 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,
%T A139336 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,
%U A139336 266,504,784,1016,1107,1015,777,483,231,77,15,1,1,7,13,14,62,2,130,28,
               173
%V A139336 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,
%W A139336 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,
%X A139336 266,-504,784,-1016,1107,-1015,777,-483,231,-77,15,-1,1,7,13,-14,-62,-2,
               130,28,-173
%N A139336 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}}.
%C A139336 Row sums are: ( different from the cyclotomics)
%C A139336 {1, -1, 1, -5, 1, -21, 1, -85, 1, -341}:
%C A139336 Plotting the roots shows that they are all cyclotomic like of complex 
               radius one:
%C A139336 w = Flatten[Table[Table[{Re[x], Im[x]} /. NSolve[p[x, n] == 0, x][[m]], 
               {m, 1, n}], {n, 1, 10}], 1];
%C A139336 ListPlot[w]
%D A139336 Weisstein, Eric W. "Sylvester Matrix." http://mathworld.wolfram.com/SylvesterMatrix.html
%D A139336 Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 232
%D A139336 Brendan Hassett, Introduction to algebraic Geometry,Cambridge University 
               Press.
%F A139336 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))).
%e A139336 {1, -2, 3, -1},
%e A139336 {1, 1, -2, -1, 1, -1},
%e A139336 {1, -4,10, -16, 19, -15, 7, -1},
%e A139336 {1, 3, -1, -8, 1, 9, -4, -6, 1, -1},
%e A139336 {1, -6, 21, -50, 90, -126,141, -125, 85, -40, 11, -1},
%e A139336 {1, 5, 4, -15, -19, 24, 29, -29, -20, 20, -6, -15, 1, -1},
%e A139336 {1, -8, 36, -112,266, -504, 784, -1016, 1107, -1015, 777, -483, 231, 
               -77, 15, -1},
%e A139336 {1, 7, 13, -14, -62, -2, 130, 28, -173, -19, 154, -28, -98, 14, -8, -28, 
               1, -1},
%e A139336 {1, -10, 55, -210, 615, -1452, 2850, -4740, 6765, -8350, 8953, -8349, 
               6756, -4704,2766, -1326, 489, -126, 19, -1},
%e A139336 {1, 9, 26,3, -116, -131, 229, 400, -305, -655, 364, 681, -417, -441, 
               342, 54, -285, -45, -10, -45, 1, -1}
%t A139336 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]
%Y A139336 Sequence in context: A122587 A086195 A086197 this_sequence A100619 A094006 
               A140188
%Y A139336 Adjacent sequences: A139333 A139334 A139335 this_sequence A139337 A139338 
               A139339
%K A139336 uned,sign
%O A139336 1,2
%A A139336 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 
               09 2008