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A202971 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202970; by antidiagonals. 2
1, -1, 1, -11, 1, 1, -30, 57, -1, 1, -53, 338, -224, 1, 1, -80, 992, -2600, 752, -1, 1, -111, 2171, -11803, 15614, -2304, 1, 1, -146, 4039, -35908, 105335, -79786, 6665, -1, 1, -185, 6776, -87154, 434244, -770624, 362449, -18595 (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..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

Top of the array:

1...-1

1...-11...1

1...-30...57....-1

1...-53...338...-224...1

MATHEMATICA

f[k_] := -2 + Fibonacci[k + 3]

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. A202970, A202605.

Sequence in context: A202767 A060270 A202678 * A202675 A176198 A202870

Adjacent sequences:  A202968 A202969 A202970 * A202972 A202973 A202974

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Dec 27 2011

STATUS

approved

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Last modified November 22 03:43 EST 2019. Contains 329388 sequences. (Running on oeis4.)