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A203999 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).

%I A203999 #7 Jul 12 2012 00:39:54
%S A203999 1,-1,-4,-6,1,7,27,17,-1,-10,-60,-99,-36,1,13,105,279,269,65,-1,-16,
%T A203999 -162,-593,-944,-609,-106,1,19,231,1077,2405,2610,1218,161,-1,-22,
%U A203999 -312,-1767,-5092,-7865,-6264,-2226,-232,1,25,405,2699,9541
%N A203999 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).
%C A203999 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 A203999 (For references regarding interlacing roots, see A202605.)
%e A203999 Top of the array:
%e A203999 1....-1
%e A203999 -4....-6.....1
%e A203999 7.... 27....17...-1
%e A203999 -10...-60...-99...-36...1
%t A203999 f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 1];
%t A203999 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203999 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A203999 Flatten[Table[f[i, n + 1 - i],
%t A203999 {n, 1, 12}, {i, 1, n}]]    (* A203998 *)
%t A203999 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203999 c[n_] := CoefficientList[p[n], x]
%t A203999 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203999 Table[c[n], {n, 1, 12}]
%t A203999 Flatten[%]   (* A203999 *)
%t A203999 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203999 Cf. A203998, A202605.
%K A203999 tabl,sign
%O A203999 1,3
%A A203999 _Clark Kimberling_, Jan 09 2012