A203002 - OEIS

%I #10 Oct 02 2017 09:58:52

%S 1,-1,1,-3,1,1,-14,21,-1,1,-29,162,-120,1,1,-48,540,-1736,844,-1,1,

%T -71,1267,-8091,17022,-5664,1,1,-98,2475,-24908,105503,-158690,39045,

%U -1,1,-129,4312,-60994,408508,-1250056,1416673

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203001; by antidiagonals.

%C 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).

%H S.-G. Hwang, <a href="http://matrix.skku.ac.kr/Series-E/Monthly-E.pdf">Cauchy's interlace theorem for eigenvalues of Hermitian matrices</a>, American Mathematical Monthly 111 (2004) 157-159.

%H A. Mercer and P. Mercer, <a href="http://dx.doi.org/10.1155/S016117120000257X">Cauchy's interlace theorem and lower bounds for the spectral radius</a>, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

%e Top of the array:

%e 1...-1

%e 1...-3....1

%e 1...-14...21....-1

%e 1...-29...162...-120...1

%t f[k_] := Fibonacci[k]^2;

%t U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];

%t L[n_] := Transpose[U[n]];

%t F[n_] := CharacteristicPolynomial[L[n].U[n], x];

%t c[n_] := CoefficientList[F[n], x]

%t TableForm[Flatten[Table[F[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A203001, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 27 2011