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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by gcd(L(i+1), L(j+1)), where L=A000032 (Lucas numbers).
3

%I #10 Aug 02 2019 04:12:37

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

%T 5280,-17696,14368,-4668,682,-44,1,147840,-506048,426568,-147856,

%U 24262,-1952,73,-1,6800640,-23573888,20317360

%N Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by gcd(L(i+1), L(j+1)), where L=A000032 (Lucas 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 2, -4, 1;

%e 6, -16, 8, -1;

%e 36, -108, 69, -15, 1;

%t u[n_] := LucasL[n]

%t f[i_, j_] := GCD[u[i], u[j]];

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

%t TableForm[m[8]] (* 8 X 8 principal submatrix *)

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

%t {n, 1, 15}, {i, 1, n}]] (* A204114 *)

%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[%] (* A204115 *)

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

%Y Cf. A204114, A202605, A204016.

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 11 2012