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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.
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%I #12 Oct 02 2017 09:36:46

%S 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 70,-21,1,1,-13,66,-165,210,-126,28,-1,1,-15,91,-286,495,-462,210,-36,

%U 1,1,-17,120,-455,1001,-1287,924,-330,45,-1,1,-19,153

%N 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 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 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 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 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 The 1st principal submatrix (ps) of A087062 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{1,1},{1,2}}, with p(2)=1-3x+x^2 and zero-set {0.381..., 2.618...}.

%e ...

%e 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 ...

%e Top of the array:

%e 1...-1

%e 1...-3....1

%e 1...-5....6....-1

%e 1...-7...15...-10....1

%e 1...-9...28...-35...15...-1

%t U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {k, 1, n}]];

%t L[n_] := Transpose[U[n]];

%t F[n_] := CharacteristicPolynomial[L[n].U[n], x];

%t c[n_] := CoefficientList[F[n], x]

%t TableForm[Flatten[Table[F[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]

%t TableForm[Table[c[n], {n, 1, 10}]]

%t Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)

%t Table[(F[k] /. x -> 2), {k, 1, 30}] (* A087168 *)

%Y Cf. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 22 2011