%I
%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 nth principal submatrix of (i+j), as in A003057.
%C Let p(n)=p(n,x) be the characteristic polynomial of the nth 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
