cp's OEIS Frontend

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