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A204025 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of gcd(i,j) (A003989). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

REFERENCES

(For references regarding interlacing roots, see A202605.)

LINKS

Table of n, a(n) for n=1..47.

EXAMPLE

Top of the array:

  1,  -1;

  1,  -3,   1;

  2,  -8,   6,  -1;

  4, -20,  26, -10,   1;

MATHEMATICA

f[i_, j_] := GCD[i, j]

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

TableForm[m[6]] (* 6 X 6 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}]]

CROSSREFS

Cf. A003989, A202605, A204016.

Sequence in context: A052914 A131671 A060750 * A204126 A204113 A204128

Adjacent sequences:  A204022 A204023 A204024 * A204026 A204027 A204028

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

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Last modified February 19 10:03 EST 2020. Contains 332041 sequences. (Running on oeis4.)