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A204113 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers).

%I A204113 #12 Aug 02 2019 04:12:45
%S A204113 1,-1,1,-3,1,2,-8,6,-1,8,-36,35,-11,1,48,-232,274,-116,19,-1,576,
%T A204113 -2880,3620,-1728,358,-32,1,10368,-52992,70632,-37192,8906,-1016,53,
%U A204113 -1,331776,-1716480,2354112,-1294352,332812,-42924,2805
%N A204113 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers).
%C A204113 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 A204113 (For references regarding interlacing roots, see A202605.)
%e A204113 Top of the array:
%e A204113   1,  -1;
%e A204113   1,  -3,   1;
%e A204113   2,  -8,   6,  -1;
%e A204113   8, -36,  35, -11,   1;
%t A204113 u[n_] := Fibonacci[n + 1]
%t A204113 f[i_, j_] := GCD[u[i], u[j]];
%t A204113 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204113 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204113 Flatten[Table[f[i, n + 1 - i],
%t A204113   {n, 1, 15}, {i, 1, n}]]    (* A204112 *)
%t A204113 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204113 c[n_] := CoefficientList[p[n], x]
%t A204113 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204113 Table[c[n], {n, 1, 12}]
%t A204113 Flatten[%]                   (* A204113 *)
%t A204113 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204113 Cf. A204112, A202605, A204016.
%K A204113 tabl,sign
%O A204113 1,4
%A A204113 _Clark Kimberling_, Jan 11 2012