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A204167 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.

%I A204167 #20 Nov 30 2021 03:48:21
%S A204167 1,-1,-2,-3,1,1,6,6,-1,0,-4,-16,-10,1,0,0,15,32,15,-1,0,0,0,-36,-60,
%T A204167 -21,1,0,0,0,0,84,100,28,-1,0,0,0,0,0,-160,-160,-36,1,0,0,0,0,0,0,300,
%U A204167 240,45,-1,0,0,0,0,0,0,0,-500,-350
%N A204167 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.
%C A204167 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 A204167 (For references regarding interlacing roots, see A202605.)
%e A204167 Top of the array:
%e A204167    1,  -1
%e A204167   -2,  -3,   1
%e A204167    1,   6,   6,  -1
%e A204167    0,  -4, -16, -10,   1
%t A204167 f[i_, j_] := Ceiling[(i + j)/2];
%t A204167 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204167 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204167 Flatten[Table[f[i, n + 1 - i],
%t A204167   {n, 1, 15}, {i, 1, n}]]  (* A204166 *)
%t A204167 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204167 c[n_] := CoefficientList[p[n], x]
%t A204167 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204167 Table[c[n], {n, 1, 12}]
%t A204167 Flatten[%]                 (* A204167 *)
%t A204167 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204167 Cf. A204166, A202605, A204016.
%K A204167 tabf,sign
%O A204167 1,3
%A A204167 _Clark Kimberling_, Jan 12 2012
%E A204167 Definition corrected by _Georg Fischer_, Nov 29 2021