A204027 - OEIS

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

%S 1,-1,1,-3,1,1,-5,6,-1,2,-12,21,-11,1,6,-40,86,-70,19,-1,30,-212,508,

%T -510,214,-32,1,240,-1756,4482,-5056,2646,-614,53,-1,3120,-23308,

%U 61748,-74480,44002,-12764,1703,-87,1,65520,-495708,1343084

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of M (as in A204026), given by min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers).

%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 1....-3....1

%e 1....-5....6....-1

%e 2....-12...21...-11....1

%t f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[6]] (* 6x6 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (* A204026 *)

%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[%]                 (* A204027 *)

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

%Y Cf. A204026, A202605, A204016.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Jan 11 2012