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A204170 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i*j), as in A003991. 2
1, -1, 0, -5, 1, 0, 0, 14, -1, 0, 0, 0, -30, 1, 0, 0, 0, 0, 55, -1, 0, 0, 0, 0, 0, -91, 1, 0, 0, 0, 0, 0, 0, 140, -1, 0, 0, 0, 0, 0, 0, 0, -204, 1, 0, 0, 0, 0, 0, 0, 0, 0, 285, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -385, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 506, -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.

p(n,x)=x^n+((-1)^n)*s(n)*x^n-1, where s=A000330 (square pyramidal numbers).

REFERENCES

(For references regarding interlacing roots, see A202605.)

LINKS

Table of n, a(n) for n=1..77.

EXAMPLE

Top of the array:

1...-1

0...-5....1

0....0....14...-1

0....0....0....-30...1

MATHEMATICA

f[i_, j_] := i*j;

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}]]  (* A003991 *)

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[%]                (* A204170 *)

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

CROSSREFS

Cf. A003991, A202605, A204016.

Sequence in context: A058177 A204619 A228077 * A283784 A281563 A293087

Adjacent sequences:  A204167 A204168 A204169 * A204171 A204172 A204173

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified January 28 07:07 EST 2020. Contains 331317 sequences. (Running on oeis4.)