A099448 A Chebyshev transform of A030191 associated to the knot 7_6.

1, 5, 19, 65, 216, 715, 2369, 7855, 26051, 86400, 286549, 950345, 3151831, 10453085, 34667784, 114976135, 381319781, 1264651795, 4194233399, 13910227200, 46133441401, 153002131805, 507433471819, 1682909416265, 5581389996216

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

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)).

Links

• Dror Bar-Natan, The Rolfsen Knot Table
• Index entries for linear recurrences with constant coefficients, signature (5,-7,5,-1).

Crossrefs

Cf. A030191, A099449.

Programs

Mathematica

CoefficientList[Series[(1+x^2)/(1-5x+7x^2-5x^3+x^4),{x,0,30}],x] (* or *)
LinearRecurrence[{5,-7,5,-1},{1,5,19,65},30] (* Harvey P. Dale, Nov 27 2013 *)

Formula

G.f.: (1+x^2)/(1-5*x+7*x^2-5*x^3+x^4).

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)).

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

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)*(1+(-1)^k)/2.