A202877 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202875; by antidiagonals.

1, -1, 1, -6, 1, 1, -11, 27, -1, 1, -17, 84, -97, 1, 1, -23, 177, -497, 311, -1, 1, -29, 306, -1405, 2546, -925, 1, 1, -35, 471, -3034, 9375, -11628, 2628, -1, 1, -41, 672, -5599, 24817, -55080, 48875, -7247, 1, 1, -47, 909, -9316, 54164

Offset: 1

Author

Clark Kimberling, Dec 26 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...-11...27...-1
1...-17...84...-97...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. A202876, A202605.

Programs

Mathematica

f[k_] := -1 + Fibonacci[k + 2]
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}]]