A204119 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(prime(i), prime(j)) (A204118).

2, -1, 5, -5, 1, 22, -28, 10, -1, 140, -204, 95, -17, 1, 1448, -2272, 1210, -278, 28, -1, 17856, -29680, 17444, -4732, 637, -41, 1, 291456, -504832, 317576, -96040, 15386, -1328, 58, -1, 5338368, -9577728, 6373968

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:
    2,   -1;
    5,   -5,    1;
   22,  -28,   10,   -1;
  140, -204,   95,  -17,    1;
		

References

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

Crossrefs

Cf. A204118, A202605, A204016.

Programs

Mathematica

f[i_, j_] := GCD[Prime[i], Prime[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}]]  (* A204118 *)
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[%]                 (* A204119 *)
TableForm[Table[c[n], {n, 1, 10}]]