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A203004 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203003; by antidiagonals.

%I A203004 #13 Oct 02 2017 09:58:59
%S A203004 1,-1,1,-18,1,1,-84,116,-1,1,-439,1221,-839,1,1,-2475,10435,-13855,
%T A203004 5658,-1,1,-14312,81690,-165715,138669,-39038,1,1,-83270,601411,
%U A203004 -1661956,2164099,-1292751,266899,-1,1,-485157
%N A203004 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203003; by antidiagonals.
%C A203004 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 A203004 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 A203004 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 A203004 Top of the array:
%e A203004 1...-1
%e A203004 1...-18....1
%e A203004 1...-84....116....-1
%e A203004 1...-439...1221...-839...1
%t A203004 f[k_] := Fibonacci[k + 1]^2;
%t A203004 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A203004 L[n_] := Transpose[U[n]];
%t A203004 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203004 c[n_] := CoefficientList[F[n], x]
%t A203004 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A203004 Table[c[n], {n, 1, 12}]
%t A203004 Flatten[%]
%t A203004 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203004 Cf. A203003, A202605.
%K A203004 tabl,sign
%O A203004 1,4
%A A203004 _Clark Kimberling_, Dec 27 2011