A204025 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of gcd(i,j) (A003989).

1, -1, 1, -3, 1, 2, -8, 6, -1, 4, -20, 26, -10, 1, 16, -88, 134, -72, 15, -1, 32, -240, 496, -408, 143, -21, 1, 192, -1504, 3352, -3112, 1344, -284, 28, -1, 768, -6400, 16320, -18496, 10508, -3108, 480, -36, 1, 4608, -39936, 109952

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;
  4, -20,  26, -10,   1;
		

References

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

Crossrefs

Cf. A003989, A202605, A204016.

Programs

Mathematica

f[i_, j_] := GCD[i, j]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]]    (* A003989 *)
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[%]                 (* A204025 *)
TableForm[Table[c[n], {n, 1, 10}]]