A202870 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals.

1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -162, 299, -99, 1, 1, -567, 1675, -1324, 225, -1, 1, -1872, 8316, -11315, 5292, -432, 1, 1, -5881, 40254, -79457, 60782, -16458, 760, -1, 1, -17990, 182413, -490520, 543130, -260498, 45424, -1232

Offset

1,4

Comments

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).

Links

Table of n, a(n) for n=1..43.

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.

A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Example

The 1st principal submatrix (ps) of A202869 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
...
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...}.
...
Top of the array:
1...-1
1...-11....1
1...-46....37....-1
1...-162...299...-99...1

Mathematica

f[k_] := Floor[k*GoldenRatio];
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]  (* A202870 as a sequence *)
TableForm[Table[c[n], {n, 1, 10}]]  (* A202870 as a matrix *)

Crossrefs

Cf. A202869, A202605.

Sequence in context: A202971 A202675 A176198 * A202872 A144440 A157209

Adjacent sequences: A202867 A202868 A202869 * A202871 A202872 A202873

Keyword

tabl,sign

Author

Clark Kimberling, Dec 26 2011

Status

approved