cp's OEIS Frontend

A204128 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

%I A204128 #6 Jul 12 2012 00:39:58
%S A204128 1,-1,1,-3,1,2,-8,6,-1,8,-36,35,-11,1,56,-268,295,-119,19,-1,672,
%T A204128 -3328,3914,-1786,361,-32,1,13440,-67904,82936,-40496,9237,-1027,53,
%U A204128 -1,443520,-2267712,2832024,-1437872,350799,-43879,2822
%N A204128 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).
%C A204128 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 A204128 (For references regarding interlacing roots, see A202605.)
%e A204128 Top of the array:
%e A204128 1....-1
%e A204128 1....-3.....1
%e A204128 2....-8.....6....-1
%e A204128 8....-36....35...-11...1
%t A204128 f[i_, j_] := 1; f[i_, i_] := Fibonacci[i + 1];
%t A204128 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204128 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204128 Flatten[Table[f[i, n + 1 - i],
%t A204128   {n, 1, 15}, {i, 1, n}]]  (* A204127 *)
%t A204128 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204128 c[n_] := CoefficientList[p[n], x]
%t A204128 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204128 Table[c[n], {n, 1, 12}]
%t A204128 Flatten[%]                 (* A204128 *)
%t A204128 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204128 Cf. A204127, A202605, A204016.
%K A204128 tabl,sign
%O A204128 1,4
%A A204128 _Clark Kimberling_, Jan 11 2012