A099446 - OEIS

%I #16 Jan 15 2025 01:45:37

%S 1,3,5,3,-8,-27,-37,-3,103,240,233,-189,-1115,-1941,-1048,3405,10747,

%T 14013,-433,-42528,-94127,-85053,88325,450387,748504,343605,-1448869,

%U -4269507,-5281865,811728,17484857,36819843,30752293

%N A Chebyshev transform of A057083.

%C The denominator is a parameterization of the Alexander polynomial for the knot 6_3. The g.f. is the image of the g.f. of A057083 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 (3,-5,3,-1).

%F G.f.: (1+x^2)/(1-3*x+5*x^2-3*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)*(-3)^j*3^(n-2*k-2*j)).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A057083(n-2*k).

%F a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A057083(k)/2.

%F a(n) = Sum_{k=0..n} A099447(n-k)*(1+(-1)^k)/2.

%Y Cf. A057083, A099447.

%K easy,sign

%O 0,2

%A _Paul Barry_, Oct 16 2004