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A203002 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203001; by antidiagonals. 2
1, -1, 1, -3, 1, 1, -14, 21, -1, 1, -29, 162, -120, 1, 1, -48, 540, -1736, 844, -1, 1, -71, 1267, -8091, 17022, -5664, 1, 1, -98, 2475, -24908, 105503, -158690, 39045, -1, 1, -129, 4312, -60994, 408508, -1250056, 1416673 (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..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

Top of the array:

1...-1

1...-3....1

1...-14...21....-1

1...-29...162...-120...1

MATHEMATICA

f[k_] := Fibonacci[k]^2;

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. A203001, A202605.

Sequence in context: A174690 A156869 A153090 * A073483 A006956 A072285

Adjacent sequences:  A202999 A203000 A203001 * A203003 A203004 A203005

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Dec 27 2011

STATUS

approved

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)