A202767 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202873; by antidiagonals.

1, -1, 1, -11, 1, 1, -25, 70, -1, 1, -39, 335, -354, 1, 1, -53, 796, -3243, 1599, -1, 1, -67, 1453, -11396, 25654, -6813, 1, 1, -81, 2306, -27557, 129202, -177146, 28156, -1, 1, -95, 3355, -54470, 407695, -1239902, 1111042, -114524

Offset: 1

Author

Clark Kimberling, Dec 26 2011

Keywords

• tabl
• sign

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 are interlace the zeros of p(n+1).

Examples

			The 1st principal submatrix (ps) of A202873 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,3},{3,7}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
...
The 3rd ps is {{1,3,7},{3,10,24},{7,24,59}}, with p(3)=1-25x+70x^2-x^3 and zero-set {0.045..., 0.312..., 69.641...}.
...
Top of the array:
1...-1
1...-11....1
1...-25...70.....-1
1...-39..335...-354...1
		

Links

• 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.

Crossrefs

Cf. A202873.

Programs

Mathematica

f[k_] := -1 + 2^k;
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[%] (* A202767 *)
TableForm[Table[c[n], {n, 1, 10}]]