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%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
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