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A204168 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j), as in A003057.

%I A204168 #6 Jul 12 2012 00:39:59
%S A204168 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 A204168 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 A204168 0,0,0,0,0,0,0,0,-825,-110,1
%N A204168 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j), as in A003057.
%C A204168 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 A204168 (For references regarding interlacing roots, see A202605.)
%e A204168 Top of the array:
%e A204168  2....-1
%e A204168 -1....-6.....1
%e A204168  0.....6.....12....-1
%e A204168  0.....0....-20....-20...1
%t A204168 f[i_, j_] := i + j;
%t A204168 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204168 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204168 Flatten[Table[f[i, n + 1 - i],
%t A204168   {n, 1, 15}, {i, 1, n}]]  (* A003057 *)
%t A204168 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204168 c[n_] := CoefficientList[p[n], x]
%t A204168 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204168 Table[c[n], {n, 1, 12}]
%t A204168 Flatten[%]                 (* A204168 *)
%t A204168 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204168 Cf. A003057, A202605, A204016.
%K A204168 tabl,sign
%O A204168 1,1
%A A204168 _Clark Kimberling_, Jan 12 2012