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%I #9 Aug 02 2019 04:12:08
%S 1,-1,1,-3,1,2,-8,6,-1,4,-20,26,-10,1,16,-88,134,-72,15,-1,32,-240,
%T 496,-408,143,-21,1,192,-1504,3352,-3112,1344,-284,28,-1,768,-6400,
%U 16320,-18496,10508,-3108,480,-36,1,4608,-39936,109952
%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of gcd(i,j) (A003989).
%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 1, -3, 1;
%e 2, -8, 6, -1;
%e 4, -20, 26, -10, 1;
%t f[i_, j_] := GCD[i, j]
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[6]] (* 6 X 6 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 15}, {i, 1, n}]] (* A003989 *)
%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[%] (* A204025 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A003989, A202605, A204016.
%K tabl,sign
%O 1,4
%A _Clark Kimberling_, Jan 11 2012
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