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A204184 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181.

%I A204184 #6 Jul 12 2012 00:39:59
%S A204184 1,-1,-2,0,1,-1,3,1,-1,2,-2,-5,0,1,1,-5,-2,6,1,-1,-2,4,9,-4,-8,0,1,-1,
%T A204184 7,3,-15,-3,9,1,-1,2,-6,-13,12,21,-6,-11,0,1,1,-9,-4,28,6,-30,-4,12,1,
%U A204184 -1,-2,8,17,-24,-40,24,38,-8,-14,0,1,-1,11,5
%N A204184 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181.
%C A204184 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 A204184 (For references regarding interlacing roots, see A202605.)
%e A204184 Top of the array:
%e A204184  1..-1
%e A204184  2...0...1
%e A204184 -1...3...1..-1
%e A204184  2..-2..-5...0..1
%t A204184 f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;
%t A204184 f[i_, i_] := (-1)^(i - 1);
%t A204184 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204184 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204184 Flatten[Table[f[i, n + 1 - i],
%t A204184   {n, 1, 15}, {i, 1, n}]]  (* A204183 *)
%t A204184 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204184 c[n_] := CoefficientList[p[n], x]
%t A204184 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204184 Table[c[n], {n, 1, 12}]
%t A204184 Flatten[%]                 (* A204184 *)
%t A204184 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204184 Cf. A204183, A202605, A204016.
%K A204184 tabl,sign
%O A204184 1,3
%A A204184 _Clark Kimberling_, Jan 12 2012