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A204122

Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^(i-1), 2^(j-1)) (A144464).

1, -1, 1, -3, 1, 2, -8, 7, -1, 8, -36, 43, -15, 1, 64, -304, 414, -198, 31, -1, 1024, -4992, 7224, -3960, 849, -63, 1, 32768, -161792, 241088, -140864, 34674, -3516, 127, -1, 2097152, -10420224, 15752192, -9492480, 2493640, -290412

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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..41.

EXAMPLE

Top of the array:

1, -1;

1, -3, 1;

2, -8, 7, -1;

8, -36, 43, -15, 1;

MATHEMATICA

f[i_, j_] := GCD[2^(i - 1), 2^(j - 1)];

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

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

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

{n, 1, 15}, {i, 1, n}]] (* A144464 *)

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[%] (* A204122 *)

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

CROSSREFS

Cf. A144464, A202605, A204016.

Sequence in context: A266272 A201677 A272536 * A201657 A279384 A394164

Adjacent sequences: A204119 A204120 A204121 * A204123 A204124 A204125

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

URL: https://oeis.org/A204122

⇱ A204122 - OEIS