cp's OEIS Frontend

A014019 Inverse of 10th cyclotomic polynomial.

%I A014019 #54 Dec 18 2023 10:16:30
%S A014019 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,
%T A014019 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,
%U A014019 0,0,0,-1,-1,0,0,0,1,1,0,0,0,-1,-1,0,0,0,1
%N A014019 Inverse of 10th cyclotomic polynomial.
%C A014019 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
%C A014019 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
%C A014019 Period 10: repeat [1,1,0,0,0,-1,-1,0,0,0]. - _Wesley Ivan Hurt_, Jun 24 2015
%C A014019 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
%H A014019 Vincenzo Librandi, <a href="/A014019/b014019.txt">Table of n, a(n) for n = 0..1000</a>
%H A014019 Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>
%H A014019 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).
%H A014019 <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>
%F A014019 G.f.: 1/(1 - x + x^2 - x^3 + x^4). - _Paul Barry_, Oct 16 2004
%F A014019 a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4), n > 4. - _Wesley Ivan Hurt_, Jun 24 2015
%p A014019 with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);
%t A014019 CoefficientList[Series[1/Cyclotomic[10, x], {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 03 2014 *)
%o A014019 (PARI) Vec(1/polcyclo(10)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 24 2014
%o A014019 (Magma) &cat[[1,1,0,0,0,-1,-1,0,0,0]: n in [0..15]]; // _Vincenzo Librandi_, Apr 03 2014
%Y A014019 Cf. A000045, A049310, A099443, A281861.
%K A014019 sign,easy
%O A014019 0,1
%A A014019 _Simon Plouffe_