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A204161 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (3i-2 if i=j and = 0 otherwise), as in A204160. 3
1, -1, 3, -5, 1, 18, -36, 12, -1, 162, -360, 153, -22, 1, 1944, -4644, 2295, -435, 35, -1, 29160, -73548, 40419, -9135, 990, -51, 1, 524880, -1382184, 823284, -210924, 27720, -1953, 70, -1, 11022480 (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..36.

EXAMPLE

Top of the array:

1.....-1

3.....-5.....1

18....-36....12....-1

162...-360...153...-22...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}]]    (* A204160 *)

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[%]                   (* A204161 *)

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

CROSSREFS

Cf. A204160, A202605, A204016.

Sequence in context: A204020 A265649 A216520 * A278968 A220110 A240752

Adjacent sequences:  A204158 A204159 A204160 * A204162 A204163 A204164

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified March 22 12:46 EDT 2019. Contains 321421 sequences. (Running on oeis4.)