%I #7 Jul 12 2012 00:39:58

%S 1,-1,1,-3,1,2,-8,6,-1,6,-28,29,-10,1,24,-124,155,-75,15,-1,120,-668,

%T 949,-565,160,-21,1,720,-4248,6636,-4564,1610,-301,28,-1,5040,-31176,

%U 52464,-40208,16569,-3892,518,-36,1,40320,-259488,463956

%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 6....-28....29...-10...1

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

%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}]] (* A204125 *)

%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[%] (* A204126 *)

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

%Y Cf. A204125, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 11 2012