A204124 - OEIS

%I #15 Feb 13 2023 03:05:31

%S 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 -102,-6,1,0,4,8,-128,73,161,7,-1,1,-4,-76,126,322,-164,-245,-8,1,1,

%U -33,63,285,-295,-629,277,351,9,-1,-4,22,121,-256,-722,662

%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)) (A144464).

%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   -3,  -2,   1;

%e   -1,  11,   3,  -1;

%e    6,  -6, -29,  -4,   1;

%t f[i_, j_] := Max[Floor[i/j], Floor[j/i]];

%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}]]  (* A204123 *)

%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[%]                 (* A204124 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204123, A202605, A204016.

%K tabf,sign

%O 1,3

%A _Clark Kimberling_, Jan 11 2012