A014019 Inverse of 10th cyclotomic polynomial.

1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1

Offset: 0

Author

Simon Plouffe

Keywords

• sign
• easy

Comments

The g.f. is the image of the g.f. of Fib(n+1) by the transform A(x) -> (1/(1 + x^2)^2)*A(x/(1+x^2)). The denominator is associated with the knots 4_1 and 5_1 by their Alexander and Jones polynomials respectively. - Paul Barry, Oct 16 2004

For n >= 1, a(n) is the determinant of an n X n Toeplitz matrix M satisfying: M(i,j) = 1 if -1 <= j - i <= 3 and 0 otherwise. - Dmitry Efimov, Jun 23 2015

Period 10: repeat [1,1,0,0,0,-1,-1,0,0,0]. - Wesley Ivan Hurt, Jun 24 2015

This is the Riordan transform of {A000045(n+1)}, n >= 0, with the Riordan matrix A049310 (Chebyshev S) of the Bell type. See the first comment by Paul Barry. - Wolfdieter Lang, Feb 18 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Dror Bar-Natan, The Rolfsen Knot Table
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).
• Index to sequences related to inverse of cyclotomic polynomials

Crossrefs

Cf. A000045, A049310, A099443, A281861.

Programs

Magma

&cat[[1,1,0,0,0,-1,-1,0,0,0]: n in [0..15]]; // Vincenzo Librandi, Apr 03 2014

Maple

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);

Mathematica

CoefficientList[Series[1/Cyclotomic[10, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)

PARI

Vec(1/polcyclo(10)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

Formula

G.f.: 1/(1 - x + x^2 - x^3 + x^4). - Paul Barry, Oct 16 2004

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4), n > 4. - Wesley Ivan Hurt, Jun 24 2015