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%I #5 Mar 30 2012 18:58:08
%S 1,1,1,1,-1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0,-1,0,0,1,1,0,0,0,0,
%T 0,0,1,1,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,-1,0,0,0,0,1,1,
%U 0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0
%N Symmetric matrix based on f(i,j) defined by f(i,1)=f(1,j)=1; f(i,i)= (-1)^(i-1); f(i,j)=0 otherwise; by antidiagonals.
%C A204183 represents the matrix M given by f(i,j) for i>=1 and j>=1. See A204184 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
%e Northwest corner:
%e 1...1...1...1...1...1
%e 1..-1...0...0...0...0
%e 1...0...1...0...0...0
%e 1...0...0..-1...0...0
%e 1...0...0...0...1...0
%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. A204184, A204016, A202453.
%K sign,tabl
%O 1
%A _Clark Kimberling_, Jan 12 2012
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