A099454 A Chebyshev transform of A099453 associated to the knot 8_12.

1, 7, 37, 175, 792, 3521, 15539, 68369, 300431, 1319472, 5793745, 25437727, 111681277, 490315231, 2152620360, 9450575729, 41490490763, 182153978153, 799702876895, 3510901281888, 15413758929889, 67670362004791, 297090274041301, 1304302623454159, 5726223576745848

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 8_12. The g.f. is the image of the g.f. of A099453 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Links

• Stefano Spezia, Table of n, a(n) for n = 0..1500
• Dror Bar-Natan, The Rolfsen Knot Table.
• Index entries for linear recurrences with constant coefficients, signature (7,-13,7,-1).

Crossrefs

Cf. A099453, A099455.

Programs

Mathematica

CoefficientList[Series[(1+x^2)/(1-7x+13x^2-7x^3+x^4),{x,0,24}],x] (* Stefano Spezia, May 13 2024 *)

Formula

G.f.: (1+x^2)/(1-7x+13x^2-7x^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)*(-11)^j*7^(n-2*k-2*j).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099453(n-2*k).

a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099453(k)/2.

a(n) = Sum_{k=0..n} A099455(n-k)*binomial(1, k/2)*(1+(-1)^k)/2.