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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).
3
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
1,3
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:
1, -1;
2, -4, 1;
12, -28, 11, -1;
144, -360, 182, -26, 1;
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}]]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved