A099448 - OEIS

%I #17 Jan 15 2025 01:45:32

%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)/(1-5*x+7*x^2-5*x^3+x^4).

%F 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*5^(n-2*k-2*j)).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A030191(n-2*k).

%F a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A030191(k)/2.

%F a(n) = Sum_{k=0..n} A099449(n-k)*(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 *)

%Y Cf. A030191, A099449.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 16 2004