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A204013 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).

%I A204013 #6 Jul 12 2012 00:39:54
%S A204013 1,-1,1,-6,1,0,-10,15,-1,-4,-8,40,-28,1,-16,24,56,-110,45,-1,-48,160,
%T A204013 -72,-224,245,-66,1,-128,608,-880,120,672,-476,91,-1,-320,1920,-4160,
%U A204013 3520,0,-1680,840,-120,1,-768,5504,-15360,20384
%N A204013 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).
%C A204013 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 A204013 (For references regarding interlacing roots, see A202605.)
%e A204013 Top of the array:
%e A204013  1....-1
%e A204013  1....-6....1
%e A204013  0....-10...15....-1
%e A204013 -4....-8....40....-28....1
%t A204013 f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];
%t A204013 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204013 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204013 Flatten[Table[f[i, n + 1 - i],
%t A204013 {n, 1, 12}, {i, 1, n}]]   (* A204012 *)
%t A204013 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204013 c[n_] := CoefficientList[p[n], x]
%t A204013 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204013 Table[c[n], {n, 1, 12}]
%t A204013 Flatten[%]  (* A204013 *)
%t A204013 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204013 Cf. A204012, A202605.
%K A204013 tabl,sign
%O A204013 1,4
%A A204013 _Clark Kimberling_, Jan 10 2012