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A204172 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171. 3
1, -1, 0, -1, 1, -1, 1, 2, -1, 0, 1, -1, -2, 1, 1, -1, -4, 3, 3, -1, 0, -1, 1, 4, -3, -3, 1, -1, 1, 6, -5, -10, 6, 4, -1, 0, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 0, -1, 1, 8, -7, -21, 15, 20, -10, -5, 1, -1, 1, 10, -9, -36, 28, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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..72.

EXAMPLE

Top of the array:

1....-1

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

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

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

MATHEMATICA

f[i_, j_] := If[Mod[Max[i, j], 2] == 1, 1, 0]

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}]]  (* A204171 *)

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[%]                 (* A204172 *)

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

CROSSREFS

Cf. A204171, A202605, A204016.

Sequence in context: A054070 A293461 A255482 * A126304 A280522 A324796

Adjacent sequences:  A204169 A204170 A204171 * A204173 A204174 A204175

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified January 24 01:05 EST 2020. Contains 331178 sequences. (Running on oeis4.)