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A204167 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164. 3
1, -1, -2, -3, 1, 1, 6, 6, -1, 0, -4, -16, -10, 1, 0, 0, 15, 32, 15, -1, 0, 0, 0, -36, -60, -21, 1, 0, 0, 0, 0, 84, 100, 28, -1, 0, 0, 0, 0, 0, -160, -160, -36, 1, 0, 0, 0, 0, 0, 0, 300, 240, 45, -1, 0, 0, 0, 0, 0, 0, 0, -500, -350 (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..63.

EXAMPLE

Top of the array:

1....-1

-2....-3.....1

1.....6.....6....-1

0....-4....-16...-10...1

MATHEMATICA

f[i_, j_] := Ceiling[(i + j)/2];

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}]]  (* A204166 *)

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[%]                 (* A204167 *)

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

CROSSREFS

Cf. A204166, A202605, A204016.

Sequence in context: A182933 A068348 A308290 * A217897 A135900 A173272

Adjacent sequences:  A204164 A204165 A204166 * A204168 A204169 A204170

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified February 27 09:44 EST 2020. Contains 332301 sequences. (Running on oeis4.)