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A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

%I A204165 #6 Jul 12 2012 00:39:59
%S A204165 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 A204165 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 A204165 -1,0,0,0,0,0,0,0,500,-75,-55,1,0,0,0,0,0,0,0,0,-825,130
%N A204165 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 A204165 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 A204165 (For references regarding interlacing roots, see A202605.)
%e A204165 Top of the array:
%e A204165  1....-1
%e A204165  1....-3.....1
%e A204165 -1....-2.....6....-1
%e A204165  0.....4.....4....-10...1
%t A204165 f[i_, j_] := Floor[(i + j)/2];
%t A204165 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204165 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204165 Flatten[Table[f[i, n + 1 - i],
%t A204165   {n, 1, 15}, {i, 1, n}]]  (* A204164 *)
%t A204165 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204165 c[n_] := CoefficientList[p[n], x]
%t A204165 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204165 Table[c[n], {n, 1, 12}]
%t A204165 Flatten[%]                 (* A204165 *)
%t A204165 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204165 Cf. A204164, A202605, A204016.
%K A204165 tabl,sign
%O A204165 1,4
%A A204165 _Clark Kimberling_, Jan 12 2012