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.

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

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