A204163 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (floor[(i+1)/2] if i=j and = 0 otherwise), as in A204162.

1, -1, 0, -2, 1, 0, -2, 4, -1, 0, -2, 7, -6, 1, 0, -4, 17, -21, 9, -1, 0, -8, 40, -64, 43, -12, 1, 0, -24, 132, -244, 206, -85, 16, -1, 0, -72, 432, -904, 913, -492, 142, -20, 1, 0, -288, 1836, -4180, 4749, -3025, 1118, -234, 25, -1, 0

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
0....-2....1
0....-2....4....-1
0....-4....17...-21...9...1
		

References

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

Crossrefs

Cf. A204162, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/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}]]  (* A204162 *)
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[%]                (* A204163 *)
TableForm[Table[c[n], {n, 1, 10}]]