A203005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.

1, -1, 1, -6, 1, 1, -15, 47, -1, 1, -40, 270, -488, 1, 1, -165, 1738, -5866, 5829, -1, 1, -1074, 15695, -80060, 156495, -74674, 1, 1, -9039, 181581, -1360515, 4552003, -5997165, 997295, -1, 1, -86700, 2566036, -28081556

Offset: 1

Author

Clark Kimberling, Dec 27 2011

Keywords

• tabl
• sign

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Examples

			Top of the array:
1...-1
1...-6....1
1...-15...47....-1
1...-40...270...-488...1
		

Links

• S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
• A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Crossrefs

Cf. A115255, A202605.

Programs

Mathematica

f[k_] := Binomial[2 k - 2, k - 1];
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]