cp's OEIS Frontend

A204115 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).

%I A204115 #10 Aug 02 2019 04:12:37
%S A204115 1,-1,2,-4,1,6,-16,8,-1,36,-108,69,-15,1,360,-1152,834,-230,26,-1,
%T A204115 5280,-17696,14368,-4668,682,-44,1,147840,-506048,426568,-147856,
%U A204115 24262,-1952,73,-1,6800640,-23573888,20317360
%N A204115 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 A204115 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 A204115 (For references regarding interlacing roots, see A202605.)
%e A204115 Top of the array:
%e A204115    1,   -1;
%e A204115    2,   -4,    1;
%e A204115    6,  -16,    8,   -1;
%e A204115   36, -108,   69,  -15,    1;
%t A204115 u[n_] := LucasL[n]
%t A204115 f[i_, j_] := GCD[u[i], u[j]];
%t A204115 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204115 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204115 Flatten[Table[f[i, n + 1 - i],
%t A204115   {n, 1, 15}, {i, 1, n}]]  (* A204114 *)
%t A204115 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204115 c[n_] := CoefficientList[p[n], x]
%t A204115 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204115 Table[c[n], {n, 1, 12}]
%t A204115 Flatten[%]                 (* A204115 *)
%t A204115 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204115 Cf. A204114, A202605, A204016.
%K A204115 tabl,sign
%O A204115 1,3
%A A204115 _Clark Kimberling_, Jan 11 2012