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A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

%I A203989 #16 Jan 30 2013 16:37:33
%S A203989 1,-1,-2,-3,1,3,11,6,-1,-4,-23,-35,-10,1,5,39,98,85,15,-1,-6,-59,-207,
%T A203989 -308,-175,-21,1,7,83,374,795,798,322,28,-1,-8,-111,-611,-1694,-2475,
%U A203989 -1806,-546,-36,1,9,143,930,3185,6149,6633,3696,870,45
%N A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).
%C A203989 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.
%C A203989 The characteristic polynomial seems be the recurrence relation given by p(n,x) = -x * p(n-1,x) + n * (-1)^(n-1) * sum_{i=0..n-1} x^i * binomial(2n-i-2,i). - _Enrique Pérez Herrero_, Jan 29 2013
%D A203989 (For references regarding interlacing roots, see A202605.)
%e A203989 Top of the array:
%e A203989 1... -1
%e A203989 -2... -3.... 1
%e A203989 3.... 11... 6... -1
%e A203989 -4... -23.. -35.. -10...1
%e A203989 5.... 39... 98... 85...15.. -1
%t A203989 f[i_, j_] := Max[i, j];
%t A203989 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203989 TableForm[m[6]] (* 6th principal submatrix *)
%t A203989 Flatten[Table[f[i, n + 1 - i],
%t A203989 {n, 1, 12}, {i, 1, n}]]  (* A051125 *)
%t A203989 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203989 c[n_] := CoefficientList[p[n], x]
%t A203989 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203989 Table[c[n], {n, 1, 12}]
%t A203989 Flatten[%]              (* A203989 *)
%t A203989 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203989 Cf. A051125, A202605.
%Y A203989 Cf. A181983, A076756.
%K A203989 tabl,sign
%O A203989 1,3
%A A203989 _Clark Kimberling_, Jan 09 2012