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A204111 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=GCD(i+1,j+1) (A204030). 3
2, -1, 5, -5, 1, 10, -20, 9, -1, 44, -100, 62, -14, 1, 104, -328, 330, -128, 20, -1, 656, -2208, 2476, -1176, 263, -27, 1, 2624, -10144, 13992, -8880, 2804, -452, 35, -1, 15744, -66112, 102384, -75760, 29512, -6336, 744, -44, 1, 67584 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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..45.

EXAMPLE

Top of the array:

2....-1

5....-5.....1

10...-20....9....-1

44...-100...62...-14...1

MATHEMATICA

f[i_, j_] := GCD[i + 1, j + 1];

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

TableForm[m[8]] (* 8x8 principal submatrix *)

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

  {n, 1, 15}, {i, 1, n}]]  (* A204030 *)

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[%]         (* A204111 *)

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

CROSSREFS

Cf. A204030, A202605, A204016.

Sequence in context: A209695 A033282 A126350 * A079502 A209164 A209148

Adjacent sequences:  A204108 A204109 A204110 * A204112 A204113 A204114

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

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Last modified June 20 22:32 EDT 2018. Contains 305615 sequences. (Running on oeis4.)