A204029 - OEIS

%I #6 Jul 12 2012 00:39:58

%S 1,-1,3,-5,1,9,-21,12,-1,27,-81,75,-22,1,81,-297,378,-195,35,-1,243,

%T -1053,1701,-1260,420,-51,1,729,-3645,7128,-6885,3402,-798,70,-1,2187,

%U -12393,28431,-33858,22275,-7938,1386,-92,1,6561

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=min(3i-2,3j-2) (A204028).

%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....-5....1

%e 9....-21...12...-1

%e 27...-81...75...-22....-11

%t f[i_, j_] := Min[3 i - 2, 3 j - 2];

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[6]] (* 6x6 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (* A204028 *)

%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[%]                 (* A204029 *)

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

%Y Cf. A204028, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 11 2012