A099846 -id:A099846 - cp's OEIS frontend

A099854 A Chebyshev transform of A048739 related to the knot 8_5.

1, 3, 7, 14, 26, 48, 90, 170, 321, 605, 1139, 2144, 4037, 7603, 14319, 26966, 50782, 95632, 180094, 339154, 638697, 1202797, 2265111, 4265664, 8033113, 15127987, 28489079, 53650734, 101035250, 190269936, 358317010, 674783850, 1270755313

Offset: 0

Paul Barry, Oct 27 2004

The g.f. is a transformation of the g.f. 1/((1-x)*(1-2*x-x^2)) of A048739 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 of the knot 8_5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,5,-4,3,-1).

Cf. A010892, A048739, A099846, A112575.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^2)^2/((1-x+x^2)*(1-2*x+x^2-2*x^3+x^4)) )); // G. C. Greubel, Apr 20 2023

LinearRecurrence[{3,-4,5,-4,3,-1}, {1,3,7,14,26,48}, 51] (* G. C. Greubel, Apr 20 2023 *)

def A099854_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^2/((1-x+x^2)*(1-2*x+x^2-2*x^3+x^4)) ).list()
A099854_list(50) # G. C. Greubel, Apr 20 2023

G.f.: (1 + x^2)^2/(1 - 3*x + 4*x^2 - 5*x^3 + 4*x^4 - 3*x^5 + x^6).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A048739(n-2*k).
a(n) = Sum_{k=0..n} A099846(n-k)*binomial(2, k/2)*(1+(-1)^k)/2.
a(n) = (1/2)*(3*A112575(n+1) + A112575(n) + 3*A112575(n-1) - A010892(n)). - G. C. Greubel, Apr 20 2023

cp's OEIS Frontend

A099854 A Chebyshev transform of A048739 related to the knot 8_5.

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