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A204169 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j-1), as in A002024. 2
1, -1, -1, -4, 1, 0, 6, 9, -1, 0, 0, -20, -16, 1, 0, 0, 0, 50, 25, -1, 0, 0, 0, 0, -105, -36, 1, 0, 0, 0, 0, 0, 196, 49, -1, 0, 0, 0, 0, 0, 0, -336, -64, 1, 0, 0, 0, 0, 0, 0, 0, 540, 81, -1, 0, 0, 0, 0, 0, 0, 0, 0, -825, -100, 1 (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..65.

EXAMPLE

Top of the array:

2....-1

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

0.....6.....9....-1

0.....0....-20...-16...1

MATHEMATICA

f[i_, j_] := i + j - 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}]]  (* A002024 *)

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[%]                (* A204169 *)

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

CROSSREFS

Cf. A002024, A202605, A204016.

Sequence in context: A303126 A292404 A060196 * A255331 A296794 A119305

Adjacent sequences:  A204166 A204167 A204168 * A204170 A204171 A204172

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)