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A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

%I A202672 #12 Oct 02 2017 09:36:46
%S A202672 1,-1,1,-3,1,1,-5,6,-1,1,-7,15,-10,1,1,-9,28,-35,15,-1,1,-11,45,-84,
%T A202672 70,-21,1,1,-13,66,-165,210,-126,28,-1,1,-15,91,-286,495,-462,210,-36,
%U A202672 1,1,-17,120,-455,1001,-1287,924,-330,45,-1,1,-19,153
%N A202672 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.
%C A202672 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A087062.  The zeros of p(n) are positive, and they interlace the zeros of p(n+1).
%C A202672 Closely related to A076756; however, for example, successive rows of A076756 are (1,-3,1), (-1,5,-6,1), compared to rows (1,-3,1), (1,-5,6,-1) of A202672.
%H A202672 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 A202672 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 A202672 The 1st principal submatrix (ps) of A087062 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
%e A202672 ...
%e A202672 The 2nd ps is {{1,1},{1,2}}, with p(2)=1-3x+x^2 and zero-set {0.381..., 2.618...}.
%e A202672 ...
%e A202672 The 3rd ps is {{1,1,1},{1,2,2},{1,2,3}}, with p(3)=1-5x+6x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
%e A202672 ...
%e A202672 Top of the array:
%e A202672 1...-1
%e A202672 1...-3....1
%e A202672 1...-5....6....-1
%e A202672 1...-7...15...-10....1
%e A202672 1...-9...28...-35...15...-1
%t A202672 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {k, 1, n}]];
%t A202672 L[n_] := Transpose[U[n]];
%t A202672 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202672 c[n_] := CoefficientList[F[n], x]
%t A202672 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202672 Table[c[n], {n, 1, 12}]
%t A202672 Flatten[%]
%t A202672 TableForm[Table[c[n], {n, 1, 10}]]
%t A202672 Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)
%t A202672 Table[(F[k] /. x -> 2), {k, 1, 30}]  (* A087168 *)
%Y A202672 Cf. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.
%K A202672 tabl,sign
%O A202672 1,4
%A A202672 _Clark Kimberling_, Dec 22 2011