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A204003 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j,i+2j} (A204002).

%I A204003 #7 Jul 12 2012 00:39:54
%S A204003 3,-1,2,-9,1,1,-9,18,-1,0,-5,25,-30,1,-1,3,14,-55,45,-1,-2,15,-27,-28,
%T A204003 105,-63,1,-3,31,-110,135,42,-182,84,-1,-4,51,-247,550,-495,-42,294,
%U A204003 -108,1,-5,75,-450,1365,-2145,1485,0,-450,135,-1,-6,103
%N A204003 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j,i+2j} (A204002).
%C A204003 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 A204003 (For references regarding interlacing roots, see A202605.)
%e A204003 Top of the array:
%e A204003 3...-1
%e A204003 2...-9.....1
%e A204003 1...-9....18...-1
%e A204003 0...-5....25...-30...1
%t A204003 f[i_, j_] := Min[2 i + j, 2 j + i];
%t A204003 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204003 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204003 Flatten[Table[f[i, n + 1 - i],
%t A204003 {n, 1, 12}, {i, 1, n}]]    (* A204002 *)
%t A204003 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204003 c[n_] := CoefficientList[p[n], x]
%t A204003 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204003 Table[c[n], {n, 1, 12}]
%t A204003 Flatten[%]                 (* A204003 *)
%t A204003 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204003 Cf. A204002, A202605.
%K A204003 tabl,sign
%O A204003 1,1
%A A204003 _Clark Kimberling_, Jan 09 2012