A099451 A Chebyshev transform of A099450 associated to the knot 7_7.

1, 5, 17, 45, 96, 155, 119, -365, -2217, -7360, -18791, -38435, -57639, -28875, 200992, 1015075, 3179711, 7796715, 15240559, 20915840, 3218033, -103746315, -458355231, -1362884995, -3211177504, -5977952405, -7345234233, 2382397955, 51340513351, 204512766400, 579756435849

Offset

0,2

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. The g.f. is the image of the g.f. of A099450 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Links

Table of n, a(n) for n=0..30.

Dror Bar-Natan, The Rolfsen Knot Table

Index entries for linear recurrences with constant coefficients, signature (5,-9,5,-1).

Formula

G.f.: (1+x^2)/(1-5x+9x^2-5x^3+x^4).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k * Sum_{j=0..n-2k} C(n-2k-j, j)*(-7)^j*5^(n-2k-2j).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099450(n-2k).

a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099450(k)/2.

a(n) = Sum_{k even, 0<=k<=n} A099452(n-k). [corrected by Kevin Ryde, Jul 24 2022]

Crossrefs

Cf. A099450, A099452.

Sequence in context: A163424 A294102 A190969 * A174794 A133252 A299335

Adjacent sequences: A099448 A099449 A099450 * A099452 A099453 A099454

Keyword

easy,sign

Author

Paul Barry, Oct 16 2004

Status

approved