A204176 - OEIS

%I #12 May 15 2017 23:40:41

%S 0,-1,-1,-1,1,0,1,1,-1,1,1,-3,-2,1,0,-1,-1,3,2,-1,-1,-1,5,4,-6,-3,1,0,

%T 1,1,-5,-4,6,3,-1,1,1,-7,-6,15,10,-10,-4,1,0,-1,-1,7,6,-15,-10,10,4,

%U -1,-1,-1,9,8,-28,-21,35,20,-15,-5,1,0,1,1,-9,-8,28,21

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is even, and 0 otherwise) as in A204175.

%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 0....-1

%e -1....-1.....1

%e 0.....1.....1....-1

%e 1.....1....-3....-2...1

%t f[i_, j_] := If[Mod[Max[i, j], 2] == 0, 1, 0]

%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}]]  (* A204175 *)

%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[%]                 (* A204176 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204175, A202605, A204016.

%K tabf,sign

%O 1,12

%A _Clark Kimberling_, Jan 12 2012