A204027 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of M (as in A204026), given by min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers).

1, -1, 1, -3, 1, 1, -5, 6, -1, 2, -12, 21, -11, 1, 6, -40, 86, -70, 19, -1, 30, -212, 508, -510, 214, -32, 1, 240, -1756, 4482, -5056, 2646, -614, 53, -1, 3120, -23308, 61748, -74480, 44002, -12764, 1703, -87, 1, 65520, -495708, 1343084

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
1....-5....6....-1
2....-12...21...-11....1
		

References

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

Crossrefs

Cf. A204026, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204026 *)
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[%]                 (* A204027 *)
TableForm[Table[c[n], {n, 1, 10}]]