A204130 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).

1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 6120, -20304, 15726, -4890, 693, -44, 1, 171360, -580752, 467724, -155524, 24797, -1963, 73, -1, 7882560, -27057312, 22300752, -7709504

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
2....-4.....1
6....-16....8....-1
36...-108...69...-15...1
		

References

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

Crossrefs

Cf. A204129, A202605, A204016.

Programs

Mathematica

f[i_, j_] := 1; f[i_, i_] := LucasL[i];
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}]]  (* A204129 *)
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[%]                 (* A204130 *)
TableForm[Table[c[n], {n, 1, 10}]]