A204183 - OEIS

%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