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A202671 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202670 based on A000290 (the squares); by antidiagonals.

%I A202671 #13 Oct 02 2017 09:36:49
%S A202671 1,-1,1,-18,1,1,-84,116,-1,1,-214,1707,-470,1,1,-408,9430,-17896,1449,
%T A202671 -1,1,-666,31877,-196046,124782,-3724,1,1,-988,81720,-1120768,2530948,
%U A202671 -656400,8400,-1,1,-1374,175727,-4386774,23536143
%N A202671 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202670 based on A000290 (the squares); by antidiagonals.
%C A202671 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 A202671 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 A202671 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 A202671 The 1st principal submatrix (ps) of A202670 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202671 ...
%e A202671 The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.556..., 17.944...}.
%e A202671 ...
%e A202671 The 3rd ps is {{1,4,9},{4,17,40},{9,40,98}}, with p(3)=1-84x+116x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
%e A202671 ...
%e A202671 Top of the array:
%e A202671 1...-1
%e A202671 1...-18..  ..1
%e A202671 1...-84... 116.....-1
%e A202671 1...-214...1707..-470...1
%t A202671 f[k_] := k^2
%t A202671 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202671 L[n_] := Transpose[U[n]];
%t A202671 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202671 c[n_] := CoefficientList[F[n], x]
%t A202671 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202671 Table[c[n], {n, 1, 12}]
%t A202671 Flatten[%]
%t A202671 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202671 Cf. A202670, A000290, A202605 (the Fibonacci case).
%K A202671 tabl,sign
%O A202671 1,4
%A A202671 _Clark Kimberling_, Dec 22 2011