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A202971 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202970; by antidiagonals.

%I A202971 #11 Oct 02 2017 09:58:46
%S A202971 1,-1,1,-11,1,1,-30,57,-1,1,-53,338,-224,1,1,-80,992,-2600,752,-1,1,
%T A202971 -111,2171,-11803,15614,-2304,1,1,-146,4039,-35908,105335,-79786,6665,
%U A202971 -1,1,-185,6776,-87154,434244,-770624,362449,-18595
%N A202971 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202970; by antidiagonals.
%C A202971 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 A202971 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 A202971 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 A202971 Top of the array:
%e A202971 1...-1
%e A202971 1...-11...1
%e A202971 1...-30...57....-1
%e A202971 1...-53...338...-224...1
%t A202971 f[k_] := -2 + Fibonacci[k + 3]
%t A202971 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202971 L[n_] := Transpose[U[n]];
%t A202971 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202971 c[n_] := CoefficientList[F[n], x]
%t A202971 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202971 Table[c[n], {n, 1, 12}]
%t A202971 Flatten[%]
%t A202971 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202971 Cf. A202970, A202605.
%K A202971 tabl,sign
%O A202971 1,4
%A A202971 _Clark Kimberling_, Dec 27 2011