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A202868 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115216; by antidiagonals.

%I A202868 #13 Oct 02 2017 09:58:11
%S A202868 1,-1,1,-6,1,1,-11,27,-1,1,-16,78,-112,1,1,-21,154,-458,453,-1,1,-26,
%T A202868 255,-1164,2431,-1818,1,1,-31,381,-2355,7635,-12141,7279,-1,1,-36,532,
%U A202868 -4156,18390,-45660,58260,-29124,1,1,-41,708,-6692,37646,-128190
%N A202868 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115216; by antidiagonals.
%C A202868 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 A202868 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 A202868 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 A202868 The 1st principal submatrix (ps) of A115216 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202868 ...
%e A202868 The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
%e A202868 ...
%e A202868 The 3rd ps is {{1,2,4},{2,5,10},{4,10,21}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.136..., 0.276..., 2.587...}.
%e A202868 ...
%e A202868 Top of the array:
%e A202868 1...-1
%e A202868 1...-6....1
%e A202868 1...-11...27...-1
%e A202868 1...-16...78...-112...1
%t A202868 f[k_] := 2^(k - 1);
%t A202868 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202868 L[n_] := Transpose[U[n]];
%t A202868 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202868 c[n_] := CoefficientList[F[n], x]
%t A202868 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202868 Table[c[n], {n, 1, 12}]
%t A202868 Flatten[%]  (* A202868 sequence *)
%t A202868 TableForm[Table[c[n], {n, 1, 10}]]  (* A202868 array *)
%t A202868 Table[(F[k] /. x -> -1), {k, 1, 30}]   (* A154626 *)
%t A202868 Table[(F[k] /. x -> 1), {k, 1, 30}]    (* A058922 *)
%Y A202868 Cf. A115216, A202605.
%K A202868 tabl,sign
%O A202868 1,4
%A A202868 _Clark Kimberling_, Dec 26 2011