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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).
3
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
1,1
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.
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
2, -1;
5, -5, 1;
22, -28, 10, -1;
140, -204, 95, -17, 1;
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}]]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved