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A204132 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131). 3
1, -1, 2, -4, 1, 8, -20, 9, -1, 48, -136, 80, -16, 1, 384, -1184, 820, -220, 25, -1, 3840, -12608, 9784, -3160, 490, -36, 1, 46080, -158976, 134400, -49504, 9380, -952, 49, -1, 645120, -2317824, 2097024, -853440, 186704 (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..40.

EXAMPLE

Top of the array:

1....-1

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

8....-20....9...-1

48...-136...80..-16...1

MATHEMATICA

f[i_, j_] := 1; f[i_, i_] := 2*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}]]  (* A204131 *)

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[%]                 (* A204132 *)

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

CROSSREFS

Cf. A204131, A202605, A204016.

Sequence in context: A065288 A065264 A233034 * A277219 A204135 A077387

Adjacent sequences:  A204129 A204130 A204131 * A204133 A204134 A204135

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

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Last modified September 24 22:31 EDT 2017. Contains 292441 sequences.