|
%I #6 Jul 12 2012 00:39:58
%S 1,-1,2,-4,1,8,-20,9,-1,48,-136,80,-16,1,384,-1184,820,-220,25,-1,
%T 3840,-12608,9784,-3160,490,-36,1,46080,-158976,134400,-49504,9380,
%U -952,49,-1,645120,-2317824,2097024,-853440,186704
%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).
%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 and A204016 for guides to related sequences.
%D (For references regarding interlacing roots, see A202605.)
%e Top of the array:
%e 1....-1
%e 2....-4.....1
%e 8....-20....9...-1
%e 48...-136...80..-16...1
%t f[i_, j_] := 1; f[i_, i_] := 2*i - 1;
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[8]] (* 8x8 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 15}, {i, 1, n}]] (* A204131 *)
%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[%] (* A204132 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204131, A202605, A204016.
%K tabl,sign
%O 1,3
%A _Clark Kimberling_, Jan 11 2012
|
