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A204163 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.

%I A204163 #6 Jul 12 2012 00:39:58
%S A204163 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 A204163 -12,1,0,-24,132,-244,206,-85,16,-1,0,-72,432,-904,913,-492,142,-20,1,
%U A204163 0,-288,1836,-4180,4749,-3025,1118,-234,25,-1,0
%N A204163 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 A204163 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 A204163 (For references regarding interlacing roots, see A202605.)
%e A204163 Top of the array:
%e A204163 1....-1
%e A204163 0....-2....1
%e A204163 0....-2....4....-1
%e A204163 0....-4....17...-21...9...1
%t A204163 f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/2];
%t A204163 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204163 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204163 Flatten[Table[f[i, n + 1 - i],
%t A204163   {n, 1, 15}, {i, 1, n}]]  (* A204162 *)
%t A204163 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204163 c[n_] := CoefficientList[p[n], x]
%t A204163 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204163 Table[c[n], {n, 1, 12}]
%t A204163 Flatten[%]                (* A204163 *)
%t A204163 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204163 Cf. A204162, A202605, A204016.
%K A204163 tabl,sign
%O A204163 1,4
%A A204163 _Clark Kimberling_, Jan 12 2012