A139343 - OEIS

%I #19 Feb 18 2024 04:14:53

%S 1,-2,3,-1,1,-3,6,-7,5,-1,1,-4,10,-16,19,-15,7,-1,1,-5,15,-30,45,-51,

%T 44,-26,9,-1,1,-6,21,-50,90,-126,141,-125,85,-40,11,-1,1,-7,28,-77,

%U 161,-266,357,-393,356,-260,146,-57,13,-1,1,-8,36,-112,266,-504,784,-1016,1107,-1015,777,-483,231,-77,15,-1,1,-9,45,-156

%N A nonsense sequence.

%C Previous name was: A triangular sequence of polynomials of the coefficients of the characteristic polynomials of the Sylvester resultant matrices of the Cyclotomic polynomials: example matrix: for x^2+x+1 and x+1; {{1, 1, 1}, {1, 1, 0}, {0, 1, 1}}.

%C Row sums are one and so are the determinants.

%C These matrices start at 3 X 3 and jump by 2 at each step.

%F 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))).

%e {1, -2, 3, -1},

%e {1, -3, 6, -7, 5, -1},

%e {1, -4, 10, -16, 19, -15, 7, -1},

%e {1, -5, 15, -30, 45, -51, 44, -26, 9, -1},

%e {1, -6, 21, -50, 90, -126, 141, -125, 85, -40, 11, -1},

%e {1, -7, 28, -77,161, -266, 357, -393, 356, -260, 146, -57, 13, -1},

%e {1, -8, 36, -112, 266, -504, 784, -1016, 1107, -1015,777, -483, 231, -77, 15, -1},

%e {1, -9, 45, -156, 414, -882, 1554, -2304,2907, -3139, 2906, -2296, 1526, -826, 344, -100, 17, -1},

%e {1, -10, 55, -210, 615, -1452, 2850, -4740, 6765, -8350, 8953, -8349, 6756, -4704, 2766, -1326, 489, -126, 19, -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]] ]

%t p[x_, n_] := p[x.n] = Sum[x^i, {i, 0, n - 1}];

%t Table[Det[SylvesterMatrix1[p[x, n], p[x, n - 1], x]], {n, 3, 11}];

%t Table[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[ x, n - 1], x], x], {n, 3, 11}];

%t a = Table[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], x], {n, 3, 11}];

%t Flatten[a]

%K uned,tabf,sign,less

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 08 2008