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A204163 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (floor[(i+1)/2] if i=j and = 0 otherwise), as in A204162. 3
1, -1, 0, -2, 1, 0, -2, 4, -1, 0, -2, 7, -6, 1, 0, -4, 17, -21, 9, -1, 0, -8, 40, -64, 43, -12, 1, 0, -24, 132, -244, 206, -85, 16, -1, 0, -72, 432, -904, 913, -492, 142, -20, 1, 0, -288, 1836, -4180, 4749, -3025, 1118, -234, 25, -1, 0 (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..55.

EXAMPLE

Top of the array:

1....-1

0....-2....1

0....-2....4....-1

0....-4....17...-21...9...1

MATHEMATICA

f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/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}]]  (* A204162 *)

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[%]                (* A204163 *)

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

CROSSREFS

Cf. A204162, A202605, A204016.

Sequence in context: A104558 A206022 A115247 * A122542 A227341 A098542

Adjacent sequences:  A204160 A204161 A204162 * A204164 A204165 A204166

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified May 30 23:47 EDT 2020. Contains 334747 sequences. (Running on oeis4.)