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A202872 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202871; by antidiagonals.

%I A202872 #12 Oct 02 2017 09:58:26
%S A202872 1,-1,1,-11,1,1,-46,37,-1,1,-181,298,-112,1,1,-716,1784,-1368,308,-1,
%T A202872 1,-2851,9495,-11119,5286,-828,1,1,-11386,47431,-74940,55235,-18546,
%U A202872 2189,-1,1,-45521,227592,-453206,455080,-239360,61185,-5759
%N A202872 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202871; by antidiagonals.
%C A202872 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 A202872 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 A202872 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 A202872 The 1st principal submatrix (ps) of A202871 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202872 ...
%e A202872 The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
%e A202872 ...
%e A202872 The 3rd ps is {{1,3,4},{3,10,15},{4,15,26}}, with p(3)=1-46x+37x^2-x^3 and zero-set {0.022..., 1.265..., 35.712...}.
%e A202872 ...
%e A202872 Top of the array:
%e A202872 1...-1
%e A202872 1...-11....1
%e A202872 1...-46....37....-1
%e A202872 1...-181...298...-112...1
%t A202872 f[k_] := LucasL[k];
%t A202872 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202872 L[n_] := Transpose[U[n]];
%t A202872 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202872 c[n_] := CoefficientList[F[n], x]
%t A202872 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202872 Table[c[n], {n, 1, 12}]
%t A202872 Flatten[%]
%t A202872 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202872 Cf. A202871, A202605.
%K A202872 tabl,sign
%O A202872 1,4
%A A202872 _Clark Kimberling_, Dec 26 2011