cp's OEIS Frontend

A099457 A Chebyshev transform of A099456 associated to the knot 9_44.

%I A099457 #18 Jan 15 2025 03:00:27
%S A099457 1,4,10,16,9,-40,-169,-376,-490,36,2239,7120,13441,12844,-16470,
%T A099457 -109144,-283351,-448120,-229129,1196064,4879030,10675276,13561279,
%U A099457 -2161760,-65753919,-204313516,-379184950,-347399104,513198089
%N A099457 A Chebyshev transform of A099456 associated to the knot 9_44.
%C A099457 The denominator is a parameterization of the Alexander polynomial for the knot 9_44. The g.f. is the image of the g.f. of A099456 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).
%H A099457 Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>
%H A099457 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,4,-1).
%F A099457 G.f.: (1+x^2)/(1-4*x+7*x^2-4*x^3+x^4).
%F A099457 a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(Sum_{j=0..n-2*k} C(n-2*k-j, j)*(-5)^j*4^(n-2*k-2*j)).
%F A099457 a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099456(n-2*k).
%F A099457 a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099456(k)/2.
%F A099457 a(n) = Sum_{k=0..n} A099458(n-k)*(1+(-1)^k)/2.
%t A099457 LinearRecurrence[{4,-7,4,-1},{1,4,10,16},30] (* _Harvey P. Dale_, Jan 17 2024 *)
%Y A099457 Cf. A099456, A099458.
%K A099457 easy,sign
%O A099457 0,2
%A A099457 _Paul Barry_, Oct 16 2004