A204128 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 56, -268, 295, -119, 19, -1, 672, -3328, 3914, -1786, 361, -32, 1, 13440, -67904, 82936, -40496, 9237, -1027, 53, -1, 443520, -2267712, 2832024, -1437872, 350799, -43879, 2822

Offset: 1

Author

Clark Kimberling, Jan 11 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
2....-8.....6....-1
8....-36....35...-11...1
		

References

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

Crossrefs

Cf. A204127, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := Fibonacci[i + 1];
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}]]  (* A204127 *)
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[%]                 (* A204128 *)
TableForm[Table[c[n], {n, 1, 10}]]