A099460 A Chebyshev transform of A099459 associated to the knot 9_48.

1, 7, 39, 203, 1040, 5313, 27133, 138565, 707643, 3613904, 18456077, 94254531, 481354555, 2458260679, 12554250288, 64114111901, 327428500337, 1672165762785, 8539691368807, 43611901581472, 222724437852585

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 9_48. The g.f. is the image of the g.f. of A099459 under the Chebyshev transform A(x) -> (1/(1+x^2))*A(x/(1+x^2)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Dror Bar-Natan, 9 48, The Knot Atlas.
• Index entries for linear recurrences with constant coefficients, signature (7,-11,7,-1).

Crossrefs

Cf. A099459, A099461.

Programs

Magma

I:=[1,7,39,203]; [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, Nov 18 2021

Mathematica

LinearRecurrence[{7,-11,7,-1}, {1,7,39,203}, 30] (* G. C. Greubel, Nov 18 2021 *)

Sage

def A099460_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099460_list(30) # G. C. Greubel, Nov 18 2021

Formula

G.f.: (1+x^2)/(1 -7*x +11*x^2 -7*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)(-9)^j*7^(n-2*k-2*j) ).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-1)^k*A099459(n-2*k).

a(n) = (1/2)*Sum_{k=0..n} (-1)^((n-k)/2)*(1 + (-1)^(n+k))*binomial((n+k)/2, k) *A099459(k).

a(n) = Sum_{k=0..n} A099461(n-k)*binomial(1, k/2)*((1+(-1)^k)/2).