A140693 Triangle read by rows, characteristic polynomials of matrices; (n X n bisymmetric matrices in which both diagonals equal the (n-1)-th row of Pascal's triangle with the rest zeros). (n>=0, 0<=k<=ceiling(n/2)).

1, 1, -1, 1, -2, 1, -4, 4, 1, -8, 12, 1, -16, 76, -96, 1, -32, 260, -400, 1, -64, 1324, -9600, 14400, 1, -128, 4760, -50176, 82320, 1, -256, 22348, -771232, 8479744, -14049280, 1, -512, 82452, -4517424, 63576576, -109734912, 1, -1024, 371284, -57164880, 3387844800, -52272864000, 91445760000

Offset

0,5

Links

Table of n, a(n) for n=0..47.

Example

First few rows of the triangle as polynomials are:
  x^0
  x^1 -  1x^0
  x^2 -  2x^1
  x^3 -  4x^2 +    4x^1
  x^4 -  8x^3 +   12x^2
  x^5 - 16x^4 +   76x^3 -   96x^2
  x^6 - 32x^5 +  260x^4 -  400x^3
  x^7 - 64x^6 + 1324x^5 - 9600x^4 + 14400x^3
  ...
The 4x4 matrix = [1,0,0,1; 0,3,3,0; 0,3,3,0; 1,0,0,1]; (i.e., a bisymmetric matrix with (1,3,3,1) as both diagonals and the rest zeros). Charpoly = x^4 - 8x^3 + 12x^2.

Prog

(PARI) row(n) = Vec(charpoly(matrix(n, n, i, j, if (i==j, binomial(n-1, i-1), if (i+j==n+1, binomial(n-1, n-i))))), 1+ceil(n/2)); \\ Michel Marcus, Feb 08 2023

Crossrefs

Cf. A007318.

Sequence in context: A350122 A156708 A131250 * A131249 A048807 A134397

Adjacent sequences: A140690 A140691 A140692 * A140694 A140695 A140696

Keyword

sign,tabf

Author

Gary W. Adamson and Roger L. Bagula, May 23 2008

Status

approved