A204115 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by gcd(L(i+1), L(j+1)), where L=A000032 (Lucas numbers).

1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 5280, -17696, 14368, -4668, 682, -44, 1, 147840, -506048, 426568, -147856, 24262, -1952, 73, -1, 6800640, -23573888, 20317360

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. A204114, A202605, A204016.

Programs

Mathematica

u[n_] := LucasL[n]
f[i_, j_] := GCD[u[i], u[j]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204114 *)
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[%]                 (* A204115 *)
TableForm[Table[c[n], {n, 1, 10}]]