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