A204019 - OEIS

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A204019

Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).

1, -1, -3, -2, 1, 8, 14, 3, -1, -21, -64, -40, -4, 1, 40, 266, 280, 90, 5, -1, 125, -930, -1671, -896, -175, -6, 1, -2940, 854, 8600, 7228, 2352, 308, 7, -1, 35035, 37744, -27334, -50164, -24594, -5376, -504, -8, 1, -372400

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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). The least zero of p(n) is -n.

For n>1, the least zero of p(n) is exactly 1-n; the greatest, for p(1) to p(5) is represented by (1,3,5.701...,9.158...13.392...).

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..45.

EXAMPLE

Top of the array:

1....-1

-3....-2......1

8.....14.....3....-1

-21...-64....-40...-4...1

MATHEMATICA

f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];

m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

TableForm[m[6]] (* 6x6 principal submatrix *)

Flatten[Table[f[i, n + 1 - i],

{n, 1, 15}, {i, 1, n}]] (* A204018 *)

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[%] (* A204019 *)

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

CROSSREFS

Cf. A204018, A202605, A204016.

Sequence in context: A210803 A204144 A203992 * A196846 A375041 A101413

Adjacent sequences: A204016 A204017 A204018 * A204020 A204021 A204022

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved