A204176 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is even, and 0 otherwise) as in A204175.

0, -1, -1, -1, 1, 0, 1, 1, -1, 1, 1, -3, -2, 1, 0, -1, -1, 3, 2, -1, -1, -1, 5, 4, -6, -3, 1, 0, 1, 1, -5, -4, 6, 3, -1, 1, 1, -7, -6, 15, 10, -10, -4, 1, 0, -1, -1, 7, 6, -15, -10, 10, 4, -1, -1, -1, 9, 8, -28, -21, 35, 20, -15, -5, 1, 0, 1, 1, -9, -8, 28, 21

Offset

1,12

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

References

(For references regarding interlacing roots, see A202605.)

Links

Table of n, a(n) for n=1..72.

Example

Top of the array:
0....-1
-1....-1.....1
0.....1.....1....-1
1.....1....-3....-2...1

Mathematica

f[i_, j_] := If[Mod[Max[i, j], 2] == 0, 1, 0]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204175 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                 (* A204176 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204175, A202605, A204016.

Sequence in context: A056898 A204065 A275281 * A062160 A301296 A194703

Adjacent sequences: A204173 A204174 A204175 * A204177 A204178 A204179

Keyword

tabf,sign

Author

Clark Kimberling, Jan 12 2012

Status

approved