A099845 A Chebyshev transform of A090400 related to the knot 8_2.

1, 3, 8, 18, 37, 75, 152, 309, 631, 1290, 2636, 5385, 10999, 22464, 45881, 93711, 191404, 390942, 798497, 1630923, 3331144, 6803829, 13896755, 28383990, 57974032, 118411413, 241854191, 493984896, 1008959473, 2060790171

Offset

0,2

Comments

The g.f. is a transformation of the g.f. 1/(1-3*x+3*x^3) of A090400 under the Chebyshev transform G(x) -> (1/(1+x^2))*G(x/(1+x^2)). The denominator of the g.f. is a parameterization of the Alexander polynomial for the knot 8_2.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-3,3,-1).

Formula

G.f.: (1 + x^2)^2/(1 - 3*x + 3*x^2 - 3*x^3 + 3*x^4 - 3*x^5 + x^6).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A090400(n-2*k).

a(n) = Sum_{k=0..n} A099844(n-k)*binomial(2, k/2)*(1+(-1)^k)/2.

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6). - Wesley Ivan Hurt, Oct 10 2021

Mathematica

CoefficientList[Series[(1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6), {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 10 2021 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6) )); // G. C. Greubel, Apr 20 2023
(SageMath)
def A077952_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^2/(1-3*x+3*x^2-3*x^3+3*x^4-3*x^5+x^6) ).list()
A077952_list(40) # G. C. Greubel, Apr 20 2023

Crossrefs

Cf. A090400, A099844.

Sequence in context: A332706 A000234 A136376 * A036635 A000713 A261325

Adjacent sequences: A099842 A099843 A099844 * A099846 A099847 A099848

Keyword

easy,nonn

Author

Paul Barry, Oct 27 2004

Status

approved