%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