login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A202875 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202874; by antidiagonals. 3
1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -19, 69, -59, 1, 1, -27, 159, -303, 162, -1, 1, -36, 302, -943, 1149, -434, 1, 1, -46, 511, -2284, 4599, -3991, 1147, -1, 1, -57, 800, -4743, 13733, -19785, 13090, -3016, 1, 1, -69, 1184, -8867, 34141, -70945 (list; table; graph; refs; listen; history; text; internal format)
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..50.

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

Top of the array:

1...-1

1...-6....1

1...-12...20...-1

1...-19...69...-59...1

MATHEMATICA

f[k_] := Fibonacci[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. A202874, A202605.

Sequence in context: A174449 A174150 A202673 * A203956 A082105 A230073

Adjacent sequences:  A202872 A202873 A202874 * A202876 A202877 A202878

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Dec 26 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 17 03:06 EST 2019. Contains 329216 sequences. (Running on oeis4.)