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%I #6 Jul 12 2012 00:39:59
%S 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 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 -1,-2,8,17,-24,-40,24,38,-8,-14,0,1,-1,11,5
%N 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 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 2...0...1
%e -1...3...1..-1
%e 2..-2..-5...0..1
%t f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;
%t f[i_, i_] := (-1)^(i - 1);
%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}]] (* A204183 *)
%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[%] (* A204184 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204183, A202605, A204016.
%K tabl,sign
%O 1,3
%A _Clark Kimberling_, Jan 12 2012
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