A204163 - OEIS

%I #6 Jul 12 2012 00:39:58

%S 1,-1,0,-2,1,0,-2,4,-1,0,-2,7,-6,1,0,-4,17,-21,9,-1,0,-8,40,-64,43,

%T -12,1,0,-24,132,-244,206,-85,16,-1,0,-72,432,-904,913,-492,142,-20,1,

%U 0,-288,1836,-4180,4749,-3025,1118,-234,25,-1,0

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (floor[(i+1)/2] if i=j and = 0 otherwise), as in A204162.

%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 0....-2....1

%e 0....-2....4....-1

%e 0....-4....17...-21...9...1

%t f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/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}]]  (* A204162 *)

%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[%]                (* A204163 *)

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

%Y Cf. A204162, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 12 2012