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 A208680 Kashaev invariant for the (7,2)-torus knot. 6
 1, 143, 58081, 48571823, 69471000001, 151763444497103, 470164385248041121, 1960764928973430783983, 10591336845363318048877441, 71933835058256664782546056463, 599982842750416411984319126244961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Compare with A156370. For other Kashaev invariants see A002439 ((3,2)-torus knot), A208679 and A208681. LINKS Table of n, a(n) for n=1..11. P. 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(7*x) = x + 143*x^3/3! + 58081*x^5/5! + .... a(n) = (-1)^n/(4*n+4)*28^(2*n+1)*sum {k = 1..28} X(k)*B(2*n+2,k/28), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 28 given by X(n) = 0 except for X(28*n+5) = X(28*n+23) = 1 and X(28*n+9) = X(28*n+19) = -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. as continued fraction: A(x) = 1/(1 + 25*x - 6*28*x/(1 - 8*28*x/(1 + 25*x -...- n*(7*n-1)*28*x/(1 - n*(7*n+1)*28*x/(1 + 25*x - ... ))))). Also, A(x) = 1/(1 + 81*x - 8*28*x/(1 - 6*28*x/(1 + 81*x -...- n*(7*n+1)*28*x/(1 - n*(7*n-1)*28*x/(1 + 81*x - ... ))))). (End) a(n) ~ sin(Pi/7) * 2^(4*n) * 7^(2*n-1) * n^(2*n-1/2) / (Pi^(2*n-1/2) * exp(2*n)). - Vaclav Kotesovec, May 18 2017 CROSSREFS Cf. A002439 ((3,2)-torus knot), A156370, A208679, A208681. Sequence in context: A029555 A265102 A046179 * A204683 A205159 A205308 Adjacent sequences: A208677 A208678 A208679 * A208681 A208682 A208683 KEYWORD nonn,easy AUTHOR Peter Bala, Mar 01 2012 STATUS approved

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Last modified February 24 15:13 EST 2024. Contains 370305 sequences. (Running on oeis4.)