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A204184 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181. 3
1, -1, -2, 0, 1, -1, 3, 1, -1, 2, -2, -5, 0, 1, 1, -5, -2, 6, 1, -1, -2, 4, 9, -4, -8, 0, 1, -1, 7, 3, -15, -3, 9, 1, -1, 2, -6, -13, 12, 21, -6, -11, 0, 1, 1, -9, -4, 28, 6, -30, -4, 12, 1, -1, -2, 8, 17, -24, -40, 24, 38, -8, -14, 0, 1, -1, 11, 5 (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..68.

EXAMPLE

Top of the array:

1..-1

2...0...1

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

2..-2..-5...0..1

MATHEMATICA

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;

f[i_, i_] := (-1)^(i - 1);

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}]]  (* A204183 *)

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[%]                 (* A204184 *)

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

CROSSREFS

Cf. A204183, A202605, A204016.

Sequence in context: A331510 A319854 A124035 * A157897 A213910 A288002

Adjacent sequences:  A204181 A204182 A204183 * A204185 A204186 A204187

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified June 14 19:28 EDT 2021. Contains 345038 sequences. (Running on oeis4.)