A204165 - OEIS

%I #6 Jul 12 2012 00:39:59

%S 1,-1,1,-3,1,-1,-2,6,-1,0,4,4,-10,1,0,0,-15,-4,15,-1,0,0,0,36,3,-21,1,

%T 0,0,0,0,-84,4,28,-1,0,0,0,0,0,160,-16,-36,1,0,0,0,0,0,0,-300,40,45,

%U -1,0,0,0,0,0,0,0,500,-75,-55,1,0,0,0,0,0,0,0,0,-825,130

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

%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  1....-3.....1

%e -1....-2.....6....-1

%e  0.....4.....4....-10...1

%t f[i_, j_] := Floor[(i + j)/2];

%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}]]  (* A204164 *)

%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[%]                 (* A204165 *)

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

%Y Cf. A204164, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 12 2012