A208681 Kashaev's invariant for the (9,2)-torus knot.

1, 239, 160801, 222359759, 525750911041, 1898604115708079, 9723130520022672481, 67030256200148854573199, 598528825179130480174293121, 6719801498668147110144664875119, 92651189588518508157161032926540961

Offset

1,2

Comments

Compare with A156652. For other Kashaev invariants see A002439, A208679 and A208680.

From Peter Bala, Dec 25 2021: (Start)

We make the following conjectures:

1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 8, 4, 20, 13, 17, 1, 8, 4, 20, 13, 17, 1, 8, 4, 20, 13, 17, ...], which appears to be a purely periodic sequence with period 6 = (1/2)*phi(21).

2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients. (End)

Links

Table of n, a(n) for n=1..11.

Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).

Formula

E.g.f.: 1/2*sin(2*x)/cos(9*x) = x + 239*x^3/3! + 160801*x^5/5! + ....

a(n) = (-1)^n/(4*n+4)*36^(2*n+1)*sum {k = 1..36} X(k)*B(2*n+2,k/36), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 36 given by X(n) = 0 except for X(36*n+7) = X(36*n+29) = 1 and X(36*n+11) = X(36*n+25) = -1.

a(n) = 1/2*(-1)^(n+1)*L(-2*n-1,X) in terms of the associated L-series attached to the periodic arithmetical function X.

From Peter Bala, May 16 2017: (Start)

O.g.f. (with offset 0) as continued fraction: A(x) = 1/(1 + 49*x - 8*36*x/(1 - 10*36*x/(1 + 49*x -...- n*(9*n-1)*36*x/(1 - n*(9*n+1)*36*x/(1 + 49*x - ... ))))).

Also, A(x) = 1/(1 + 121*x - 10*36*x/(1 - 8*36*x/(1 + 121*x -...- n*(9*n+1)*36*x/(1 - n*(9*n-1)*36*x/(1 + 121*x - ... ))))). (End)

a(n) ~ sin(Pi/9) * 2^(4*n) * 3^(4*n-2) * n^(2*n-1/2) / (Pi^(2*n-1/2) * exp(2*n)). - Vaclav Kotesovec, May 18 2017

From Peter Bala, Dec 25 2021: (Start)

a(1) = 1, a(n) = (-4)^(n-1) - Sum_{k = 1..n} (-81)^k*C(2*n-1,2*k)*a(n-k).

a(n) == 239^(n-1) mod ( (2^8)*(3^4)*5 ). (End)

Maple

A208681 := proc(n) option remember; if n = 1 then 1; else (-4)^(n-1) - add((-81)^k*binomial(2*n-1, 2*k)*procname(n-k), k=1..n) ; end if; end proc:
seq(A208681(n), n = 1..20) # Peter Bala, Dec 25 2021

Mathematica

a[n_] := (2n-1)! SeriesCoefficient[(1/2)(Sin[2x]/ Cos[9x]), {x, 0, 2n-1}];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Sep 23 2022 *)

Crossrefs

Cf. A002439 ((3,2)-torus knot), A156652, A208679, A208680.

Sequence in context: A223788 A223830 A223974 * A221327 A221302 A072173

Adjacent sequences: A208678 A208679 A208680 * A208682 A208683 A208684

Keyword

nonn,easy

Author

Peter Bala, Mar 01 2012

Status

approved