A139336 A nonsense sequence.

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, 1, 7, 13, -14, -62, -2, 130, 28, -173

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}}.

Links

Table of n, a(n) for n=1..79.

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

{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},
{1, 7, 13, -14, -62, -2, 130, 28, -173, -19, 154, -28, -98, 14, -8, -28, 1, -1},
{1, -10, 55, -210, 615, -1452, 2850, -4740, 6765, -8350, 8953, -8349, 6756, -4704,2766, -1326, 489, -126, 19, -1},
{1, 9, 26,3, -116, -131, 229, 400, -305, -655, 364, 681, -417, -441, 342, 54, -285, -45, -10, -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]] ]
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]

Crossrefs

Sequence in context: A289623 A086195 A086197 * A252168 A100619 A211984

Adjacent sequences: A139333 A139334 A139335 * A139337 A139338 A139339

Keyword

uned,sign,tabf,less

Author

Roger L. Bagula and Gary W. Adamson, Jun 09 2008

Status

approved