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A202868
Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115216; by antidiagonals.
3
1, -1, 1, -6, 1, 1, -11, 27, -1, 1, -16, 78, -112, 1, 1, -21, 154, -458, 453, -1, 1, -26, 255, -1164, 2431, -1818, 1, 1, -31, 381, -2355, 7635, -12141, 7279, -1, 1, -36, 532, -4156, 18390, -45660, 58260, -29124, 1, 1, -41, 708, -6692, 37646, -128190
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
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 A115216 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 {{1,2,4},{2,5,10},{4,10,21}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.136..., 0.276..., 2.587...}.
...
Top of the array:
1...-1
1...-6....1
1...-11...27...-1
1...-16...78...-112...1
MATHEMATICA
f[k_] := 2^(k - 1);
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[%] (* A202868 sequence *)
TableForm[Table[c[n], {n, 1, 10}]] (* A202868 array *)
Table[(F[k] /. x -> -1), {k, 1, 30}] (* A154626 *)
Table[(F[k] /. x -> 1), {k, 1, 30}] (* A058922 *)
CROSSREFS
Sequence in context: A081579 A295707 A146772 * A202877 A174124 A174345
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Dec 26 2011
STATUS
approved