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A204134 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).

%I A204134 #6 Jul 12 2012 00:39:58
%S A204134 1,-1,1,-3,1,3,-11,7,-1,21,-83,64,-15,1,315,-1287,1074,-300,31,-1,
%T A204134 9765,-40527,35067,-10570,1287,-63,1,615195,-2572731,2265129,-707539,
%U A204134 92653,-5313,127,-1,78129765,-327967227,291222882,-92551369
%N A204134 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 A204134 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 A204134 (For references regarding interlacing roots, see A202605.)
%e A204134 Top of the array:
%e A204134 1....-1
%e A204134 1....-3.....1
%e A204134 3....-11....7....-1
%e A204134 21...-83....64...-15...1
%t A204134 f[i_, j_] := 1; f[i_, i_] := 2^(i - 1);
%t A204134 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204134 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204134 Flatten[Table[f[i, n + 1 - i],
%t A204134   {n, 1, 15}, {i, 1, n}]]  (* A204133 *)
%t A204134 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204134 c[n_] := CoefficientList[p[n], x]
%t A204134 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204134 Table[c[n], {n, 1, 12}]
%t A204134 Flatten[%]                 (* A204134 *)
%t A204134 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204134 Cf. A204133, A202605, A204016.
%K A204134 tabl,sign
%O A204134 1,4
%A A204134 _Clark Kimberling_, Jan 11 2012