A204117 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^i-1, 2^j-1) (A204116).

1, -1, 2, -4, 1, 12, -28, 11, -1, 144, -360, 182, -26, 1, 4320, -11088, 5940, -984, 57, -1, 233280, -616032, 348768, -64728, 4506, -120, 1, 29393280, -78086592, 44775936, -8554608, 636444, -19740, 247, -1, 7054387200

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;
   12,  -28,   11,   -1;
  144, -360,  182,  -26,    1;
		

References

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

Crossrefs

Cf. A204116, A202605, A204016.

Programs

Mathematica

f[i_, j_] := GCD[2^i - 1, 2^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}]]  (* A204116 *)
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[%]                 (* A204117 *)
TableForm[Table[c[n], {n, 1, 10}]]