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Suppose I know that the generating function of some sequence $(f_n)_n$ is $$ G(f_n;x) = \exp\left[-\alpha\cdot\arcsin(\beta\cdot x)+\frac{1}{2}\log\left(\gamma\sqrt{1-\delta \cdot x^2}+1-\delta \cdot x\right)\right] $$ for some constants $\alpha,\beta,\gamma,\delta$. How can one deduce the asymptotic formula of the coefficients corresponding to the above generating function?

It seems that usual techniques involving Tauberian theorem, or the "Asymptotic growth of a sequence" section in

https://en.wikipedia.org/wiki/Generating_function

are not so helpful. Is there any way to do that?

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    $\begingroup$ The standard answer for this is "check Flajolet and Sedgwick", although I don't know if that would actually be helpful in this case. $\endgroup$ – Michael Lugo Aug 17 '17 at 14:55

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