A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

1, -1, 1, -3, 1, -1, -2, 6, -1, 0, 4, 4, -10, 1, 0, 0, -15, -4, 15, -1, 0, 0, 0, 36, 3, -21, 1, 0, 0, 0, 0, -84, 4, 28, -1, 0, 0, 0, 0, 0, 160, -16, -36, 1, 0, 0, 0, 0, 0, 0, -300, 40, 45, -1, 0, 0, 0, 0, 0, 0, 0, 500, -75, -55, 1, 0, 0, 0, 0, 0, 0, 0, 0, -825, 130

Offset: 1

Author

Clark Kimberling, Jan 12 2012

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 real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

Examples

			Top of the array:
 1....-1
 1....-3.....1
-1....-2.....6....-1
 0.....4.....4....-10...1
		

References

• (For references regarding interlacing roots, see A202605.)

Crossrefs

Cf. A204164, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Floor[(i + j)/2];
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}]]  (* A204164 *)
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[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]