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Asymptotic growth of the Catalan numbers

The ordinary generating function for the Catalan numbers is G (C n; x) = 1 − 1 − 4 x 2 x. {\displaystyle G(C_{n};x)={\frac {1-{\sqrt {1-4x}}}{2x}}.}

With r = 1 / 4, α = 1, β = − 1 / 2, A (x) = 1 / 2, and B (x) = − 1 / 2, we can conclude that, for the Catalan numbers: C n ∼ B (r) r α Γ (β) n β − 1 (1 r) n = − 1 2 (1 4) 1 Γ (− 1 2) n − 1 2 − 1 (1 1 4) n = 4 n n 3 2 π. {\displaystyle C_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta)}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {-{\frac {1}{2}}}{\left({\frac {1}{4}}\right)^{1}\Gamma \left(-{\frac {1}{2}}\right)}}\,n^{-{\frac {1}{2}}-1}\left({\frac {1}{\,{\frac {1}{4}}\,}}\right)^{n}={\frac {4^{n}}{n^{\frac {3}{2}}{\sqrt {\pi }}}}.}