A202870 - OEIS

%I #12 Oct 02 2017 09:58:19

%S 1,-1,1,-11,1,1,-46,37,-1,1,-162,299,-99,1,1,-567,1675,-1324,225,-1,1,

%T -1872,8316,-11315,5292,-432,1,1,-5881,40254,-79457,60782,-16458,760,

%U -1,1,-17990,182413,-490520,543130,-260498,45424,-1232

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals.

%C 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 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 A202869 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.

%e ...

%e The 3rd ps is {{1,3,4},{3,10,15},{4,15,26}}, with p(3)=1-46x+37x^2-x^3 and zero-set {0.022..., 1.265..., 35.712...}.

%e ...

%e Top of the array:

%e 1...-1

%e 1...-11....1

%e 1...-46....37....-1

%e 1...-162...299...-99...1

%t f[k_] := Floor[k*GoldenRatio];

%t U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {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[%]  (* A202870 as a sequence *)

%t TableForm[Table[c[n], {n, 1, 10}]]  (* A202870 as a matrix *)

%Y Cf. A202869, A202605.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 26 2011