A204122 - OEIS

%I #10 Aug 02 2019 04:13:05

%S 1,-1,1,-3,1,2,-8,7,-1,8,-36,43,-15,1,64,-304,414,-198,31,-1,1024,

%T -4992,7224,-3960,849,-63,1,32768,-161792,241088,-140864,34674,-3516,

%U 127,-1,2097152,-10420224,15752192,-9492480,2493640,-290412

%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   1,  -3,   1;

%e   2,  -8,   7,  -1;

%e   8, -36,  43, -15,   1;

%t f[i_, j_] := GCD[2^(i - 1), 2^(j - 1)];

%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}]] (* A144464 *)

%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[%]              (* A204122 *)

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

%Y Cf. A144464, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 11 2012