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A202677 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); by antidiagonals.

%I A202677 #12 Oct 02 2017 09:57:51
%S A202677 1,-1,1,-18,1,1,-116,84,-1,1,-538,1347,-250,1,1,-2256,11566,-8216,585,
%T A202677 -1,1,-9158,75453,-118722,35086,-1176,1,1,-36796,426288,-1152432,
%U A202677 801084,-118656,2128,-1,1,-147378,2214919,-8910538,11175711,-4079622,339762,-3564,1,1,-589736,10915650
%N A202677 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); by antidiagonals.
%C A202677 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 A202677 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 A202677 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 A202677 The 1st principal submatrix (ps) of A202676 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202677 ...
%e A202677 The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.055..., 17.944...}.
%e A202677 ...
%e A202677 The 3rd ps is {{1,4,7},{4,17,32},{7,32,66}}, with p(3)=1-116x+84x^2-x^3 and zero-set {0.008..., 1.395..., 82.595...}.
%e A202677 ...
%e A202677 Top of the array:
%e A202677 1...-1
%e A202677 1...-18....1
%e A202677 1...-116...84.....-1
%e A202677 1...-538...1347...-250...1
%t A202677 f[k_] := 3 k - 2
%t A202677 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202677 L[n_] := Transpose[U[n]];
%t A202677 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202677 c[n_] := CoefficientList[F[n], x]
%t A202677 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202677 Table[c[n], {n, 1, 12}]
%t A202677 Flatten[%]
%t A202677 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202677 Cf. A202676, A202677, A202605.
%K A202677 tabl,sign
%O A202677 1,4
%A A202677 _Clark Kimberling_, Dec 22 2011