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A204165 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, 1, -3, 1, -1, -2, 6, -1, 0, 4, 4, -10, 1, 0, 0, -15, -4, 15, -1, 0, 0, 0, 36, 3, -21, 1, 0, 0, 0, 0, -84, 4, 28, -1, 0, 0, 0, 0, 0, 160, -16, -36, 1, 0, 0, 0, 0, 0, 0, -300, 40, 45, -1, 0, 0, 0, 0, 0, 0, 0, 500, -75, -55, 1, 0, 0, 0, 0, 0, 0, 0, 0, -825, 130 (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..75.

EXAMPLE

Top of the array:

1....-1

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

-1....-2.....6....-1

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

MATHEMATICA

f[i_, j_] := Floor[(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}]]  (* A204164 *)

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[%]                 (* A204165 *)

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

CROSSREFS

Cf. A204164, A202605, A204016.

Sequence in context: A143934 A318442 A086639 * A329473 A200702 A331599

Adjacent sequences:  A204162 A204163 A204164 * A204166 A204167 A204168

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified March 30 10:09 EDT 2020. Contains 333125 sequences. (Running on oeis4.)