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A204025
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of GCD(i,j) (A003989).
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2
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1, -1, 1, -3, 1, 2, -8, 6, -1, 4, -20, 26, -10, 1, 16, -88, 134, -72, 15, -1, 32, -240, 496, -408, 143, -21, 1, 192, -1504, 3352, -3112, 1344, -284, 28, -1, 768, -6400, 16320, -18496, 10508, -3108, 480, -36, 1, 4608, -39936, 109952
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OFFSET
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1,4
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COMMENTS
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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.
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REFERENCES
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(For references regarding interlacing roots, see A202605.)
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LINKS
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Table of n, a(n) for n=1..47.
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EXAMPLE
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Top of the array:
1....-1
1....-3....1
2....-8....6....-1
4....-20...26...-10....1
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MATHEMATICA
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f[i_, j_] := GCD[i, j]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A003989 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%] (* A204025 *)
TableForm[Table[c[n], {n, 1, 10}]]
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CROSSREFS
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Cf. A003989, A202605, A204016.
Sequence in context: A052914 A131671 A060750 * A204126 A204113 A204128
Adjacent sequences: A204022 A204023 A204024 * A204026 A204027 A204028
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KEYWORD
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tabl,sign
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AUTHOR
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Clark Kimberling, Jan 11 2012
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STATUS
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approved
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