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A204111 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(i+1, j+1) (A204030).

%I A204111 #11 Aug 02 2019 04:12:26
%S A204111 2,-1,5,-5,1,10,-20,9,-1,44,-100,62,-14,1,104,-328,330,-128,20,-1,656,
%T A204111 -2208,2476,-1176,263,-27,1,2624,-10144,13992,-8880,2804,-452,35,-1,
%U A204111 15744,-66112,102384,-75760,29512,-6336,744,-44,1,67584
%N A204111 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(i+1, j+1) (A204030).
%C A204111 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 A204111 (For references regarding interlacing roots, see A202605.)
%e A204111 Top of the array:
%e A204111    2,   -1;
%e A204111    5,   -5,    1;
%e A204111   10,  -20,    9,   -1;
%e A204111   44, -100,   62,  -14,    1;
%t A204111 f[i_, j_] := GCD[i + 1, j + 1];
%t A204111 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204111 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204111 Flatten[Table[f[i, n + 1 - i],
%t A204111   {n, 1, 15}, {i, 1, n}]]  (* A204030 *)
%t A204111 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204111 c[n_] := CoefficientList[p[n], x]
%t A204111 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204111 Table[c[n], {n, 1, 12}]
%t A204111 Flatten[%]         (* A204111 *)
%t A204111 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204111 Cf. A204030, A202605, A204016.
%K A204111 tabl,sign
%O A204111 1,1
%A A204111 _Clark Kimberling_, Jan 11 2012