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%I #6 Jul 12 2012 00:39:54
%S 1,-1,1,-2,1,0,-2,3,-1,-4,8,0,-4,1,-16,56,-56,10,5,-1,-48,224,-360,
%T 224,-35,-6,1,-128,736,-1584,1560,-672,84,7,-1,-320,2176,-5824,7744,
%U -5280,1680,-168,-8,1,-768,6016,-19200,32032,-29744
%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i-j+1,j-i+1} (A203994).
%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...-2....1
%e 0...-2....3...-1
%e -4....8....0...-4....1
%t f[i_, j_] := Min[i - j + 1, j - i + 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}]] (* A203994 *)
%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[%] (* A203995 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A203994, A202605.
%K tabl,sign
%O 1,4
%A _Clark Kimberling_, Jan 09 2012