A204128 - OEIS

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

%S 1,-1,1,-3,1,2,-8,6,-1,8,-36,35,-11,1,56,-268,295,-119,19,-1,672,

%T -3328,3914,-1786,361,-32,1,13440,-67904,82936,-40496,9237,-1027,53,

%U -1,443520,-2267712,2832024,-1437872,350799,-43879,2822

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

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

%e 8....-36....35...-11...1

%t f[i_, j_] := 1; f[i_, i_] := Fibonacci[i + 1];

%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}]]  (* A204127 *)

%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[%]                 (* A204128 *)

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

%Y Cf. A204127, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 11 2012