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

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

Offset: 1

Author

Peter Bala, Mar 01 2012

Keywords

• nonn
• easy

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

• 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).

Crossrefs

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

Programs

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 *)

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)