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%I A204124 #15 Feb 13 2023 03:05:31
%S A204124 1,-1,-3,-2,1,-1,11,3,-1,6,-6,-29,-4,1,1,-13,8,56,5,-1,-1,-6,71,-46,
%T A204124 -102,-6,1,0,4,8,-128,73,161,7,-1,1,-4,-76,126,322,-164,-245,-8,1,1,
%U A204124 -33,63,285,-295,-629,277,351,9,-1,-4,22,121,-256,-722,662
%N A204124 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)) (A144464).
%C A204124 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 A204124 (For references regarding interlacing roots, see A202605.)
%e A204124 Top of the array:
%e A204124    1,  -1;
%e A204124   -3,  -2,   1;
%e A204124   -1,  11,   3,  -1;
%e A204124    6,  -6, -29,  -4,   1;
%t A204124 f[i_, j_] := Max[Floor[i/j], Floor[j/i]];
%t A204124 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204124 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204124 Flatten[Table[f[i, n + 1 - i],
%t A204124   {n, 1, 15}, {i, 1, n}]]  (* A204123 *)
%t A204124 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204124 c[n_] := CoefficientList[p[n], x]
%t A204124 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204124 Table[c[n], {n, 1, 12}]
%t A204124 Flatten[%]                 (* A204124 *)
%t A204124 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204124 Cf. A204123, A202605, A204016.
%K A204124 tabf,sign
%O A204124 1,3
%A A204124 _Clark Kimberling_, Jan 11 2012

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