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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).
3

%I #6 Jul 12 2012 00:39:58

%S 1,-1,2,-4,1,6,-16,8,-1,36,-108,69,-15,1,360,-1152,834,-230,26,-1,

%T 6120,-20304,15726,-4890,693,-44,1,171360,-580752,467724,-155524,

%U 24797,-1963,73,-1,7882560,-27057312,22300752,-7709504

%N 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 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 2....-4.....1

%e 6....-16....8....-1

%e 36...-108...69...-15...1

%t f[i_, j_] := 1; f[i_, i_] := LucasL[i];

%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}]] (* A204129 *)

%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[%] (* A204130 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204129, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 11 2012