A099446 A Chebyshev transform of A057083.

1, 3, 5, 3, -8, -27, -37, -3, 103, 240, 233, -189, -1115, -1941, -1048, 3405, 10747, 14013, -433, -42528, -94127, -85053, 88325, 450387, 748504, 343605, -1448869, -4269507, -5281865, 811728, 17484857, 36819843, 30752293

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• sign

Comments

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)).

Links

• Dror Bar-Natan, The Rolfsen Knot Table
• Index entries for linear recurrences with constant coefficients, signature (3,-5,3,-1).

Crossrefs

Cf. A057083, A099447.

Formula

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

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

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

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