A202671 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202670 based on A000290 (the squares); by antidiagonals.

1, -1, 1, -18, 1, 1, -84, 116, -1, 1, -214, 1707, -470, 1, 1, -408, 9430, -17896, 1449, -1, 1, -666, 31877, -196046, 124782, -3724, 1, 1, -988, 81720, -1120768, 2530948, -656400, 8400, -1, 1, -1374, 175727, -4386774, 23536143

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..40.

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 A202670 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.556..., 17.944...}.
...
The 3rd ps is {{1,4,9},{4,17,40},{9,40,98}}, with p(3)=1-84x+116x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
...
Top of the array:
1...-1
1...-18..  ..1
1...-84... 116.....-1
1...-214...1707..-470...1

Mathematica

f[k_] := 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. A202670, A000290, A202605 (the Fibonacci case).

Sequence in context: A146774 A174451 A144405 * A203004 A155497 A202677

Adjacent sequences: A202668 A202669 A202670 * A202672 A202673 A202674

Keyword

tabl,sign

Author

Clark Kimberling, Dec 22 2011

Status

approved