A204168 - OEIS

%I #6 Jul 12 2012 00:39:59

%S 2,-1,-1,-6,1,0,6,12,-1,0,0,-20,-20,1,0,0,0,50,30,-1,0,0,0,0,-105,-42,

%T 1,0,0,0,0,0,196,56,-1,0,0,0,0,0,0,-336,-72,1,0,0,0,0,0,0,0,540,90,-1,

%U 0,0,0,0,0,0,0,0,-825,-110,1

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j), as in A003057.

%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  2....-1

%e -1....-6.....1

%e  0.....6.....12....-1

%e  0.....0....-20....-20...1

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

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

%t TableForm[m[8]] (* 8x8 principal submatrix *)

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

%t   {n, 1, 15}, {i, 1, n}]]  (* A003057 *)

%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[%]                 (* A204168 *)

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

%Y Cf. A003057, A202605, A204016.

%K tabl,sign

%O 1,1

%A _Clark Kimberling_, Jan 12 2012