A208730 Sequence related to Kashaev's invariant for the (5,2)-torus knot.
1, 2, 10, 104, 1870, 51632, 2027470, 107354144, 7370645950, 636754087472, 67591284235630, 8647294709864384, 1312197219579059230, 233025643830843282512
Offset: 0
Links
- K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).
Formula
Define F(q) := sum {m,n >= 0} (q^(-m*n)*product {i = 1.. m+n} (1-q^i)).
E.g.f.: F(exp(-t)) = 1 + 2*t + 10*t^2! + 104*t^3/3! + .... For the expansion of F(1-q) see A208733. F(q) also appears in a conjectural e.g.f. for A208679.
a(n) = (9/40)^n*sum {k = 0..n} binomial(n,k)*A208679(k+1)/9^k.
Conjectural S-fraction for the o.g.f.: 1/(1-2*x/(1-3*x/(1-9*x/(1-11*x/(1-...-1/2*n*(5*n-1)*x/(1-1/2*n*(5*n+1)*x/(1- ....
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