OFFSET
1,2
COMMENTS
Previous name was: 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}}.
FORMULA
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))).
EXAMPLE
Triangle begins:
{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, 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, 266, -504, 784, -1016, 1107, -1015, 777, -483, 231, -77, 15, -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]] ]
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[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[ x, n], p[x, n - 1], x], x], x], {n, 2, 8}];
Flatten[a]
CROSSREFS
KEYWORD
uned,sign,tabf,less
AUTHOR
Roger L. Bagula and Gary W. Adamson, Jun 09 2008
STATUS
approved
