A203992 - OEIS

%I #6 Jul 12 2012 00:39:54

%S 1,-1,-3,-2,1,8,14,3,-1,-20,-56,-40,-4,1,48,184,224,90,5,-1,-112,-544,

%T -936,-672,-175,-6,1,256,1504,3344,3480,1680,308,7,-1,-576,-3968,

%U -10816,-14784,-10560,-3696,-504,-8,1,1280,10112,32640,55328

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).

%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 for a guide to related sequences.

%D (For references regarding interlacing roots, see A202605.)

%e Top of the array:

%e  1... -1

%e -3... -1.... 1

%e  8.... 14... 3... -1

%e -20.. -56.. -40.. -4... 1

%t f[i_, j_] := Max[i - j + 1, j - i + 1];

%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, 12}, {i, 1, n}]]  (* A143182 in square format *)

%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[%]    (* A203992 *)

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

%Y Cf. A143182, A202605.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 09 2012