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%I #21 Feb 08 2023 12:37:40
%S 1,1,-1,1,-2,1,-4,4,1,-8,12,1,-16,76,-96,1,-32,260,-400,1,-64,1324,
%T -9600,14400,1,-128,4760,-50176,82320,1,-256,22348,-771232,8479744,
%U -14049280,1,-512,82452,-4517424,63576576,-109734912,1,-1024,371284,-57164880,3387844800,-52272864000,91445760000
%N 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)).
%e First few rows of the triangle as polynomials are:
%e x^0
%e x^1 - 1x^0
%e x^2 - 2x^1
%e x^3 - 4x^2 + 4x^1
%e x^4 - 8x^3 + 12x^2
%e x^5 - 16x^4 + 76x^3 - 96x^2
%e x^6 - 32x^5 + 260x^4 - 400x^3
%e x^7 - 64x^6 + 1324x^5 - 9600x^4 + 14400x^3
%e ...
%e 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.
%o (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
%Y Cf. A007318.
%K sign,tabf
%O 0,5
%A _Gary W. Adamson_ and _Roger L. Bagula_, May 23 2008