A099457 A Chebyshev transform of A099456 associated to the knot 9_44.

1, 4, 10, 16, 9, -40, -169, -376, -490, 36, 2239, 7120, 13441, 12844, -16470, -109144, -283351, -448120, -229129, 1196064, 4879030, 10675276, 13561279, -2161760, -65753919, -204313516, -379184950, -347399104, 513198089

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• sign

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 9_44. The g.f. is the image of the g.f. of A099456 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 (4,-7,4,-1).

Crossrefs

Cf. A099456, A099458.

Programs

Mathematica

LinearRecurrence[{4,-7,4,-1},{1,4,10,16},30] (* Harvey P. Dale, Jan 17 2024 *)

Formula

G.f.: (1+x^2)/(1-4*x+7*x^2-4*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*4^(n-2*k-2*j)).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099456(n-2*k).

a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099456(k)/2.

a(n) = Sum_{k=0..n} A099458(n-k)*(1+(-1)^k)/2.