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

%I A203005 #13 Oct 02 2017 09:59:05
%S A203005 1,-1,1,-6,1,1,-15,47,-1,1,-40,270,-488,1,1,-165,1738,-5866,5829,-1,1,
%T A203005 -1074,15695,-80060,156495,-74674,1,1,-9039,181581,-1360515,4552003,
%U A203005 -5997165,997295,-1,1,-86700,2566036,-28081556
%N A203005 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.
%C A203005 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 A203005 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 A203005 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 A203005 Top of the array:
%e A203005 1...-1
%e A203005 1...-6....1
%e A203005 1...-15...47....-1
%e A203005 1...-40...270...-488...1
%t A203005 f[k_] := Binomial[2 k - 2, k - 1];
%t A203005 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A203005 L[n_] := Transpose[U[n]];
%t A203005 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203005 c[n_] := CoefficientList[F[n], x]
%t A203005 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A203005 Table[c[n], {n, 1, 12}]
%t A203005 Flatten[%]
%t A203005 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203005 Cf. A115255, A202605.
%K A203005 tabl,sign
%O A203005 1,4
%A A203005 _Clark Kimberling_, Dec 27 2011