%I #10 Aug 02 2019 04:12:31
%S 1,-1,2,-4,1,12,-28,11,-1,144,-360,182,-26,1,4320,-11088,5940,-984,57,
%T -1,233280,-616032,348768,-64728,4506,-120,1,29393280,-78086592,
%U 44775936,-8554608,636444,-19740,247,-1,7054387200
%N 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).
%C 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.
%D (For references regarding interlacing roots, see A202605.)
%e Top of the array:
%e 1, -1;
%e 2, -4, 1;
%e 12, -28, 11, -1;
%e 144, -360, 182, -26, 1;
%t f[i_, j_] := GCD[2^i - 1, 2^j - 1];
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 15}, {i, 1, n}]] (* A204116 *)
%t p[n_] := CharacteristicPolynomial[m[n], x];
%t c[n_] := CoefficientList[p[n], x]
%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t Table[c[n], {n, 1, 12}]
%t Flatten[%] (* A204117 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204116, A202605, A204016.
%K tabl,sign
%O 1,3
%A _Clark Kimberling_, Jan 11 2012