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%I #14 Jan 17 2024 11:59:40
%S 1,4,10,16,9,-40,-169,-376,-490,36,2239,7120,13441,12844,-16470,
%T -109144,-283351,-448120,-229129,1196064,4879030,10675276,13561279,
%U -2161760,-65753919,-204313516,-379184950,-347399104,513198089
%N A Chebyshev transform of A099456 associated to the knot 9_44.
%C 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 Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,4,-1).
%F G.f.: (1+x^2)/(1-4x+7x^2-4x^3+x^4); a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0..n-2k, C(n-2k-j, j)(-5)^j*4^(n-2k-2j)}}; a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*A099456(n-2k)); a(n)=sum{k=0..n, binomial((n+k)/2, k)(-1)^((n-k)/2)(1+(-1)^(n+k))A099456(k)/2}; a(n)=sum{k=0..n, A099458(n-k)*binomial(1, k/2)(1+(-1)^k)/2}.
%t LinearRecurrence[{4,-7,4,-1},{1,4,10,16},30] (* _Harvey P. Dale_, Jan 17 2024 *)
%K easy,sign
%O 0,2
%A _Paul Barry_, Oct 16 2004
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