A204124 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)) (A144464).

1, -1, -3, -2, 1, -1, 11, 3, -1, 6, -6, -29, -4, 1, 1, -13, 8, 56, 5, -1, -1, -6, 71, -46, -102, -6, 1, 0, 4, 8, -128, 73, 161, 7, -1, 1, -4, -76, 126, 322, -164, -245, -8, 1, 1, -33, 63, 285, -295, -629, 277, 351, 9, -1, -4, 22, 121, -256, -722, 662

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• tabf
• 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;
  -3,  -2,   1;
  -1,  11,   3,  -1;
   6,  -6, -29,  -4,   1;
		

References

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

Crossrefs

Cf. A204123, A202605, A204016.

Programs

Mathematica

f[i_, j_] := Max[Floor[i/j], Floor[j/i]];
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}]]  (* A204123 *)
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[%]                 (* A204124 *)
TableForm[Table[c[n], {n, 1, 10}]]