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A202675 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202674 based on (1,3,5,7,9,...); by antidiagonals.

%I A202675 #12 Oct 02 2017 09:36:53
%S A202675 1,-1,1,-11,1,1,-37,46,-1,1,-79,367,-130,1,1,-137,1444,-2083,295,-1,1,
%T A202675 -211,4013,-13820,8518,-581,1,1,-301,9066,-58277,89402,-27966,1036,-1,
%U A202675 1,-407,17851,-186166,548591,-442118,78354
%N A202675 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202674 based on (1,3,5,7,9,...); by antidiagonals.
%C A202675 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 A202675 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 A202675 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 A202675 The 1st principal submatrix (ps) of A202674 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202675 ...
%e A202675 The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
%e A202675 ...
%e A202675 The 3rd ps is {{1,3,5},{3,10,18},{5,18,35}}, with p(3)=1-37x+46x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
%e A202675 ...
%e A202675 Top of the array:
%e A202675 1....-1
%e A202675 1...-11.....1
%e A202675 1...-37....46.....-1
%e A202675 1...-79...367...-130...1
%t A202675 f[k_] := 2 k - 1
%t A202675 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202675 L[n_] := Transpose[U[n]];
%t A202675 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202675 c[n_] := CoefficientList[F[n], x]
%t A202675 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202675 Table[c[n], {n, 1, 12}]
%t A202675 Flatten[%]
%t A202675 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202675 Cf. A202674, A202605.
%K A202675 tabl,sign
%O A202675 1,4
%A A202675 _Clark Kimberling_, Dec 22 2011