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A204007 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j-2,2j+i-2} (A204006).

%I A204007 #6 Jul 12 2012 00:39:54
%S A204007 1,-1,0,-5,1,-1,-1,12,-1,-2,7,5,-22,1,-3,19,-28,-15,35,-1,-4,35,-99,
%T A204007 84,35,-51,1,-5,55,-220,375,-210,-70,70,-1,-6,79,-403,990,-1155,462,
%U A204007 126,-92,1,-7,107,-660,2093,-3575,3069,-924,-210,117,-1
%N A204007 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j-2,2j+i-2} (A204006).
%C A204007 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 for a guide to related sequences.
%D A204007 (For references regarding interlacing roots, see A202605.)
%e A204007 Top of the array:
%e A204007  1....-1
%e A204007  0....-5....1
%e A204007 -1....-1....12....-1
%e A204007 -2.....7....5.....-22...1
%t A204007 f[i_, j_] := Min[2 i + j - 2, 2 j + i - 2];
%t A204007 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204007 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204007 Flatten[Table[f[i, n + 1 - i],
%t A204007 {n, 1, 12}, {i, 1, n}]]   (* A204006 *)
%t A204007 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204007 c[n_] := CoefficientList[p[n], x]
%t A204007 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204007 Table[c[n], {n, 1, 12}]
%t A204007 Flatten[%]                (* A204007 *)
%t A204007 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204007 Cf. A204006, A202605.
%K A204007 tabl,sign
%O A204007 1,4
%A A204007 _Clark Kimberling_, Jan 09 2012