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%I #6 Jul 12 2012 00:39:54
%S 1,-1,1,-6,1,1,-9,17,-1,1,-12,39,-36,1,1,-15,69,-119,65,-1,1,-18,107,
%T -272,294,-106,1,1,-21,153,-515,846,-630,161,-1,1,-24,207,-868,1925,
%U -2232,1218,-232,1,1,-27,269,-1351,3783,-6017,5214
%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1-1),j(i+1)-1} (A204000).
%C 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 (For references regarding interlacing roots, see A202605.)
%e Top of the array:
%e 1...-1
%e 1...-6....1
%e 1...-9....17...-1
%e 1...-12...39...-36...1
%t f[i_, j_] := Min[i (j + 1) - 1, j (i + 1) - 1];
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[6]] (* 6x6 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 12}, {i, 1, n}]] (* A204000 *)
%t p[n_] := CharacteristicPolynomial[m[n], x];
%t c[n_] := CoefficientList[p[n], x]
%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t Table[c[n], {n, 1, 12}]
%t Flatten[%] (* A204001 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204000, A202605.
%K tabl,sign
%O 1,4
%A _Clark Kimberling_, Jan 09 2012
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