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A202673 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals. 3
1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -18, 75, -50, 1, 1, -24, 166, -328, 105, -1, 1, -30, 293, -1050, 1134, -196, 1, 1, -36, 456, -2432, 5140, -3312, 336, -1, 1, -42, 655, -4690, 15471, -20814, 8514, -540, 1, 1, -48, 890, -8040, 36771, -80584 (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 of A115262 (when A115262 is formatted as a square matrix). 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

The 1st principal submatrix (ps) of A115263 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

...

The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.

...

The 3rd ps is {{1,2,3},{2,5,8},{3,8,14}}, with p(3)=1-12x+20x^2-x^3 and zero-set {0.099..., 0.516..., 19.383...}.

...

Top of the array:

1....-1

1....-6.....1

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

1...-18....75....-50....1

1...-24...166...-328..105..-1

MATHEMATICA

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[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. A115262, A202671 (based on n^2), A202605 (based on Fibonacci numbers)

Sequence in context: A174345 A174449 A174150 * A202875 A203956 A082105

Adjacent sequences:  A202670 A202671 A202672 * A202674 A202675 A202676

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Dec 22 2011

STATUS

approved

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Last modified September 20 12:05 EDT 2021. Contains 347586 sequences. (Running on oeis4.)