A202677 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); by antidiagonals.

1, -1, 1, -18, 1, 1, -116, 84, -1, 1, -538, 1347, -250, 1, 1, -2256, 11566, -8216, 585, -1, 1, -9158, 75453, -118722, 35086, -1176, 1, 1, -36796, 426288, -1152432, 801084, -118656, 2128, -1, 1, -147378, 2214919, -8910538, 11175711, -4079622, 339762, -3564, 1, 1, -589736, 10915650

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..47.

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 A202676 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 {0.055..., 17.944...}.
...
The 3rd ps is {{1,4,7},{4,17,32},{7,32,66}}, with p(3)=1-116x+84x^2-x^3 and zero-set {0.008..., 1.395..., 82.595...}.
...
Top of the array:
1...-1
1...-18....1
1...-116...84.....-1
1...-538...1347...-250...1

Mathematica

f[k_] := 3 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. A202676, A202677, A202605.

Sequence in context: A202671 A203004 A155497 * A179838 A174678 A167884

Adjacent sequences: A202674 A202675 A202676 * A202678 A202679 A202680

Keyword

tabl,sign

Author

Clark Kimberling, Dec 22 2011

Status

approved