A202675 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202674 based on (1,3,5,7,9,...); by antidiagonals.

1, -1, 1, -11, 1, 1, -37, 46, -1, 1, -79, 367, -130, 1, 1, -137, 1444, -2083, 295, -1, 1, -211, 4013, -13820, 8518, -581, 1, 1, -301, 9066, -58277, 89402, -27966, 1036, -1, 1, -407, 17851, -186166, 548591, -442118, 78354

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..42.

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 A202674 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 {{1,3,5},{3,10,18},{5,18,35}}, with p(3)=1-37x+46x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
...
Top of the array:
1....-1
1...-11.....1
1...-37....46.....-1
1...-79...367...-130...1

Mathematica

f[k_] := 2 k - 1
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[%]
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A202674, A202605.

Sequence in context: A060270 A202678 A202971 * A176198 A202870 A202872

Adjacent sequences: A202672 A202673 A202674 * A202676 A202677 A202678

Keyword

tabl,sign

Author

Clark Kimberling, Dec 22 2011

Status

approved