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A204029 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=min(3i-2,3j-2) (A204028). 3
1, -1, 3, -5, 1, 9, -21, 12, -1, 27, -81, 75, -22, 1, 81, -297, 378, -195, 35, -1, 243, -1053, 1701, -1260, 420, -51, 1, 729, -3645, 7128, -6885, 3402, -798, 70, -1, 2187, -12393, 28431, -33858, 22275, -7938, 1386, -92, 1, 6561 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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:

1....-1

3....-5....1

9....-21...12...-1

27...-81...75...-22....-11

MATHEMATICA

f[i_, j_] := Min[3 i - 2, 3 j - 2];

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}]]  (* A204028 *)

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[%]                 (* A204029 *)

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

CROSSREFS

Cf. A204028, A202605, A204016.

Sequence in context: A214728 A112752 A101035 * A026253 A259182 A138259

Adjacent sequences:  A204026 A204027 A204028 * A204030 A204031 A204032

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.)