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Determining the asymptotic growth of the coefficients of a generating function is essentially the entire subject of Analytic Combinatorics, on which see the 810-page book by Flajolet and Sedgewick available online.

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I thought that $\log(n!)$ would be $\Omega(n \log n )$, but I read somewhere that $\log(n!) = O(n\log n)$

Asymptotics for partial sum of product of binomial coefficients. 5. Asymptotics of a function involving ...

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Virtually any book or lecture notes on asymptotics should have the relevant information. Graham, Knuth, and Patashnik's Concrete Mathematics has a nice chapter on it, for instance. $\endgroup$ – Antonio Vargas

asymptotics; approximation. The Overflow Blog Stack Gives Back 2024! Linked. 7. What is the difference ...