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A204130 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).

%I A204130 #6 Jul 12 2012 00:39:58
%S A204130 1,-1,2,-4,1,6,-16,8,-1,36,-108,69,-15,1,360,-1152,834,-230,26,-1,
%T A204130 6120,-20304,15726,-4890,693,-44,1,171360,-580752,467724,-155524,
%U A204130 24797,-1963,73,-1,7882560,-27057312,22300752,-7709504
%N A204130 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).
%C A204130 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 A204130 (For references regarding interlacing roots, see A202605.)
%e A204130 Top of the array:
%e A204130 1....-1
%e A204130 2....-4.....1
%e A204130 6....-16....8....-1
%e A204130 36...-108...69...-15...1
%t A204130 f[i_, j_] := 1; f[i_, i_] := LucasL[i];
%t A204130 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204130 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204130 Flatten[Table[f[i, n + 1 - i],
%t A204130   {n, 1, 15}, {i, 1, n}]]  (* A204129 *)
%t A204130 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204130 c[n_] := CoefficientList[p[n], x]
%t A204130 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204130 Table[c[n], {n, 1, 12}]
%t A204130 Flatten[%]                 (* A204130 *)
%t A204130 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204130 Cf. A204129, A202605, A204016.
%K A204130 tabl,sign
%O A204130 1,3
%A A204130 _Clark Kimberling_, Jan 11 2012