A204121 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(prime(i+1), prime(j+1)) (A204120).

3, -1, 14, -8, 1, 92, -68, 15, -1, 968, -816, 230, -26, 1, 12096, -11248, 3740, -564, 39, -1, 199296, -198400, 73544, -13192, 1222, -56, 1, 3679488, -3877632, 1567824, -320304, 36160, -2280, 75, -1, 82607616, -91008000

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:
    3,   -1;
   14,   -8,    1;
   92,  -68,   15,   -1;
  968, -816,  230,  -26,    1;
		

References

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

Crossrefs

Cf. A204120, A202605, A204016.

Programs

Mathematica

f[i_, j_] := GCD[Prime[i + 1], Prime[j + 1]];
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}]]    (* A204120 *)
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[%]                   (* A204121 *)
TableForm[Table[c[n], {n, 1, 10}]]