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A202672 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals. 3
1, -1, 1, -3, 1, 1, -5, 6, -1, 1, -7, 15, -10, 1, 1, -9, 28, -35, 15, -1, 1, -11, 45, -84, 70, -21, 1, 1, -13, 66, -165, 210, -126, 28, -1, 1, -15, 91, -286, 495, -462, 210, -36, 1, 1, -17, 120, -455, 1001, -1287, 924, -330, 45, -1, 1, -19, 153 (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 A087062.  The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Closely related to A076756; however, for example, successive rows of A076756 are (1,-3,1), (-1,5,-6,1), compared to rows (1,-3,1), (1,-5,6,-1) of A202672.

LINKS

Table of n, a(n) for n=1..57.

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 A087062 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.

...

The 2nd ps is {{1,1},{1,2}}, with p(2)=1-3x+x^2 and zero-set {0.381..., 2.618...}.

...

The 3rd ps is {{1,1,1},{1,2,2},{1,2,3}}, with p(3)=1-5x+6x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.

...

Top of the array:

1...-1

1...-3....1

1...-5....6....-1

1...-7...15...-10....1

1...-9...28...-35...15...-1

MATHEMATICA

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {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}]]

Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)

Table[(F[k] /. x -> 2), {k, 1, 30}]  (* A087168 *)

CROSSREFS

Cf. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.

Sequence in context: A245368 A239331 A145033 * A054142 A076756 A114172

Adjacent sequences:  A202669 A202670 A202671 * A202673 A202674 A202675

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Dec 22 2011

STATUS

approved

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Last modified November 15 06:05 EST 2019. Contains 329144 sequences. (Running on oeis4.)