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A Chebyshev transform of A030191 associated to the knot 7_6.
1

%I #11 Sep 09 2015 01:28:32

%S 1,5,19,65,216,715,2369,7855,26051,86400,286549,950345,3151831,

%T 10453085,34667784,114976135,381319781,1264651795,4194233399,

%U 13910227200,46133441401,153002131805,507433471819,1682909416265,5581389996216

%N A Chebyshev transform of A030191 associated to the knot 7_6.

%C The denominator is a parameterization of the Alexander polynomial for the knot 7_6. The g.f. is the image of the g.f. of A030191 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 (5,-7,5,-1).

%F G.f.: (1+x^2)/(1-5x+7x^2-5x^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*5^(n-2k-2j)}}; a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*A030191(n-2k)); a(n)=sum{k=0..n, binomial((n+k)/2, k)(-1)^((n-k)/2)(1+(-1)^(n+k))A030191(k)/2}; a(n)=sum{k=0..n, A099449(n-k)*binomial(1, k/2)(1+(-1)^k)/2};

%t CoefficientList[Series[(1+x^2)/(1-5x+7x^2-5x^3+x^4),{x,0,30}],x] (* or *)

%t LinearRecurrence[{5,-7,5,-1},{1,5,19,65},30] (* _Harvey P. Dale_, Nov 27 2013 *)

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 16 2004