A136458 Triangle of coefficients of the characteristic polynomial of a bi-orthogonal n X n matrix: h(i,j) = If[i - j == 0, 1, If[Abs[i - j] - n/2 == 0, -1, 0]];i,j<=n; example n=4: {{1, 0, -1, 0}, {0, 1, 0, -1}, {-1, 0, 1, 0}, {0, -1, 0, 1}}.

1, 1, -1, 0, -2, 1, 1, -3, 3, -1, 0, 0, 4, -4, 1, 1, -5, 10, -10, 5, -1, 0, 0, 0, -8, 12, -6, 1, 1, -7, 21, -35, 35, -21, 7, -1, 0, 0, 0, 0, 16, -32, 24, -8, 1, 1, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 0, 0, 0, 0, -32, 80, -80, 40, -10, 1

Offset

1,5

Comments

Row sums are: {1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1}.

These matrices are related to binary digital signal processing.

References

http://www.ee.cityu.edu.hk/~eekwwong/ee40214/chapter3.pdf (dead link)

Links

Table of n, a(n) for n=1..66.

Formula

If[i - j == 0, 1, If[Abs[i - j] - n/2 == 0, -1, 0]],

Example

{1},
{1, -1},
{0, -2, 1},
{1, -3, 3, -1},
{0, 0, 4, -4, 1},
{1, -5, 10, -10, 5, -1},
{0, 0, 0, -8, 12, -6, 1},
{1, -7, 21, -35, 35, -21, 7, -1},
{0, 0, 0, 0, 16, -32, 24, -8, 1},
{1, -9, 36, -84, 126, -126, 84, -36, 9, -1},
{0, 0, 0, 0, 0, -32, 80, -80, 40, -10, 1}

Mathematica

Clear[B] B[n_] := Table[Table[If[i -j == 0, 1, If[Abs[i - j] - n/2 == 0, -1, 0]], {i, 1, n}], {j, 1, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[B[n], x], x], {n, 1, 10}]]; Flatten[a] Join[{1}, Table[Apply[Plus, CoefficientList[CharacteristicPolynomial[B[n], x], x]], {n, 1, 10}]];

Crossrefs

Sequence in context: A108482 A124750 A275865 * A048805 A204015 A370140

Adjacent sequences: A136455 A136456 A136457 * A136459 A136460 A136461

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Mar 20 2008

Status

approved