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A202767 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202873; by antidiagonals.

%I A202767 #11 Oct 02 2017 09:58:05
%S A202767 1,-1,1,-11,1,1,-25,70,-1,1,-39,335,-354,1,1,-53,796,-3243,1599,-1,1,
%T A202767 -67,1453,-11396,25654,-6813,1,1,-81,2306,-27557,129202,-177146,28156,
%U A202767 -1,1,-95,3355,-54470,407695,-1239902,1111042,-114524
%N A202767 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202873; by antidiagonals.
%C A202767 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 are interlace the zeros of p(n+1).
%H A202767 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 A202767 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 A202767 The 1st principal submatrix (ps) of A202873 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202767 ...
%e A202767 The 2nd ps is {{1,3},{3,7}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
%e A202767 ...
%e A202767 The 3rd ps is {{1,3,7},{3,10,24},{7,24,59}}, with p(3)=1-25x+70x^2-x^3 and zero-set {0.045..., 0.312..., 69.641...}.
%e A202767 ...
%e A202767 Top of the array:
%e A202767 1...-1
%e A202767 1...-11....1
%e A202767 1...-25...70.....-1
%e A202767 1...-39..335...-354...1
%t A202767 f[k_] := -1 + 2^k;
%t A202767 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202767 L[n_] := Transpose[U[n]];
%t A202767 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202767 c[n_] := CoefficientList[F[n], x]
%t A202767 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202767 Table[c[n], {n, 1, 12}]
%t A202767 Flatten[%] (* A202767 *)
%t A202767 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202767 Cf. A202873.
%K A202767 tabl,sign
%O A202767 1,4
%A A202767 _Clark Kimberling_, Dec 26 2011