Stirling's formula

Stirling's formula: [math]\displaystyle{ n! = (1+o(1)) (2\pi n)^{1/2} n^n e^{-n} }[/math].

As a consequence, one has

[math]\displaystyle{ \binom{n}{\alpha n + k} = (c(\alpha)+o(1)) \theta(\alpha)^k 2^{h(\alpha) n} \frac{1}{\sqrt{n}} }[/math]

for fixed [math]\displaystyle{ 0 \lt \alpha \lt 1 }[/math] and bounded k, where

[math]\displaystyle{ c(\alpha) := \frac{1}{\sqrt{2\pi \alpha(1-\alpha)}} }[/math]

[math]\displaystyle{ \theta(\alpha) := \frac{1-\alpha}{\alpha} }[/math]

[math]\displaystyle{ h(\alpha) := \alpha \log_2 \frac{1}{\alpha} + (1-\alpha) \log_2 \frac{1}{1-\alpha}. }[/math]

Thus for instance

[math]\displaystyle{ \binom{n}{n/2 + k} = (\sqrt{\frac{2}{\pi}} + o(1)) \frac{1}{\sqrt{n}} 2^n }[/math]

[math]\displaystyle{ \binom{n}{n/3 + k} = (\sqrt{\frac{9}{4\pi}} + o(1)) \frac{1}{\sqrt{n}} 3^n 2^{-2n/3 + k} }[/math]

etc. Note that the entropy function [math]\displaystyle{ h(\alpha) }[/math] attains a maximum at [math]\displaystyle{ \alpha=1/2 }[/math], which is consistent with the binomials [math]\displaystyle{ \binom{n}{j} }[/math] being maximized at [math]\displaystyle{ j \approx n/2 }[/math].

For trinomials, we have

[math]\displaystyle{ \frac{n!}{(\alpha n+i)! (\beta n + j)! (\gamma_n + k)!} = (c(\alpha,\beta,\gamma)+o(1)) \alpha^{-i} \beta^{-j} \gamma^{-k} 2^{h(\alpha,\beta,\gamma) n} \frac{1}{n} }[/math]

for fixed [math]\displaystyle{ 0 \lt \alpha,\beta,\gamma \lt 1 }[/math] and bounded i,j,k summing to 1 and 0 respectively, where

[math]\displaystyle{ c(\alpha,\beta,\gamma) := \frac{1}{2\pi} \alpha^{-1/2} \beta^{-1/2} \gamma^{-1/2} }[/math]

[math]\displaystyle{ h(\alpha,\beta,\gamma):= \alpha \log_2 \frac{1}{\alpha} + \beta \log_2 \frac{1}{\beta} + \gamma \log_2 \frac{1}{\gamma}. }[/math]

Thus for instance

[math]\displaystyle{ \frac{n!}{(n/3+i)! (n/3+j)! (n/3+k)!} = (\frac{3^{3/2}}{2\pi} + o(1)) \frac{1}{n} 3^n. }[/math]

Here is the Wikipedia article for the Stirling's formula.