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A203997 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1),j(i+1)} (A203996).

%I A203997 #6 Jul 12 2012 00:39:54
%S A203997 2,-1,3,-8,1,4,-19,20,-1,5,-34,69,-40,1,6,-53,160,-189,70,-1,7,-76,
%T A203997 305,-552,434,-112,1,8,-103,516,-1265,1560,-882,168,-1,9,-134,805,
%U A203997 -2496,4235,-3828,1638,-240,1,10,-169,1184,-4445,9646
%N A203997 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1),j(i+1)} (A203996).
%C A203997 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 A203997 (For references regarding interlacing roots, see A202605.)
%e A203997 Top of the array:
%e A203997 2...-1
%e A203997 3...-8.....1
%e A203997 4...-19....20....-1
%e A203997 5...-34....69....-40....1
%e A203997 6...-53....160...-189...70....-1
%e A203997 7...-76....305...-552...434...-112...1
%t A203997 f[i_, j_] := Min[i (j + 1), j (i + 1)];
%t A203997 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203997 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A203997 Flatten[Table[f[i, n + 1 - i],
%t A203997 {n, 1, 12}, {i, 1, n}]]   (* A203996 *)
%t A203997 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203997 c[n_] := CoefficientList[p[n], x]
%t A203997 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203997 Table[c[n], {n, 1, 12}]
%t A203997 Flatten[%]      (* A203997 *)
%t A203997 TableForm[Table[c[n], {n, 1, 10}]]
%Y A203997 Cf. A203996, A202605.
%K A203997 tabl,sign
%O A203997 1,1
%A A203997 _Clark Kimberling_, Jan 09 2012