|
%I #6 Jul 12 2012 00:39:58
%S 1,-1,1,-3,1,3,-11,7,-1,21,-83,64,-15,1,315,-1287,1074,-300,31,-1,
%T 9765,-40527,35067,-10570,1287,-63,1,615195,-2572731,2265129,-707539,
%U 92653,-5313,127,-1,78129765,-327967227,291222882,-92551369
%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 1....-3.....1
%e 3....-11....7....-1
%e 21...-83....64...-15...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}]] (* A204133 *)
%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[%] (* A204134 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204133, A202605, A204016.
%K tabl,sign
%O 1,4
%A _Clark Kimberling_, Jan 11 2012
|
