The half of logarithm of [url=https://en.wikipedia.org/wiki/Binomial_coefficient]binomial coefficient[/url] approaches the [url=https://en.wikipedia.org/wiki/Binary_entropy_function]binary entropy function[/url] [i]H[sub]b[/sub][/i] in base [i]e[/i][sup]2[/sup], as [i]n[/i] approaches infinity, which is very close to an arc of a perfect circle.[br][br]Binomial coefficient [math]B\binom{n}{k}[/math] = [math]C_k^n[/math] = number of k-combinations of n, and [url=https://en.wikipedia.org/wiki/Beta_function]beta function[/url] B(p, q):[br][math]B\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}=\frac{\left(n+1\right)^{-1}}{B\left(k+1,n-k+1\right)}[/math][br][br]Normalized log of binomial: ([i]x = k/n[/i])[br][math]lB_n\left(x\right)=\frac{ln\left(B\binom{n}{xn}\right)}{n}[/math][br][i][br][color=#38761d]lB[/color][/i] approaches [i][color=#980000]H[sub]b[/sub][/color][/i] in base [i]e[/i] as [i]n[/i] approaches infinity:[br][math]lim_{n\longrightarrow\infty}lB_n\left(x\right)=H_b\left(x\right)=-\left(x\cdot ln\left(x\right)+\left(1-x\right)ln\left(1-x\right)\right)[/math][br][br][color=#38761d]Half of [i]lB[/i][/color] approaches [i][color=#980000]H[sub]bs[/sub][/color][/i] ([i][color=#980000]H[sub]b[/sub][/color][/i] in base [i]e[sup]2[/sup][/i]) as [i]n[/i] approaches infinity:[br][math]lim_{n\longrightarrow\infty}\frac{1}{2}lB_n\left(x\right)=H_{bs}\left(x\right)=-\frac{1}{2}\left(x\cdot ln\left(x\right)+\left(1-x\right)ln\left(1-x\right)\right)[/math][br][br][color=#ff0000][i]H[sub]bs[/sub][/i] (in red)[/color] is very close to an arc of circle [i]C[/i] (in black):

In [url=https://en.wikipedia.org/wiki/Binomial_distribution]binomial distribution[/url] of a fair coin, [i]X[/i] ~ B([i]n[/i], p=1/2), the probability of getting exactly [i]k[/i] heads in [i]n[/i] independent [url=https://en.wikipedia.org/wiki/Bernoulli_trial]Bernoulli trials[/url] is the ratio between the number of [i]k[/i]-combinations and the total number of combinations 2[sup][i]n[/i][/sup]:[br][math]Pr\left[X=k\right]=\frac{B\binom{n}{k}}{2^n}=B_d\left(n,k\right)=B_d\left(x\right)\left[x=\frac{k}{n}\right][/math][br][br]When [i]n[/i] approaches infinity, this distribution approaches the [url=https://en.wikipedia.org/wiki/Normal_distribution]normal distribution[/url] N(μ=[i]n[/i]/2, σ[sup]2[/sup]=[i]n[/i]/4), and its probability mass function B[sub]d[/sub]([i]x[/i]), normalized by scaling both axes by √[i]n[/i] and centering to 1/2, approaches the probability mass function φ([i]x[/i]| μ=0, σ=1/2) of the fair normal distribution N(μ=0, σ[sup]2[/sup]=1/4):[br][math]lim_{n\rightarrow\infty}\sqrt{n}B_d\left(\frac{x}{\sqrt{n}}+\frac{1}{2}\right)=\text{φ}\left(x\mid\mu=0,\sigma=\frac{1}{2}\right)=\sqrt{\frac{2}{\pi}}e^{-2x^2}[/math][br]

Derivatives of the binary entropy function [i]H[sub]b[/sub][/i] is related to the [url=https://en.wikipedia.org/wiki/Logistic_map]logistic map[/url] [i]x[/i][i]⋅(1-x)[/i]: [br][br][math]dH\left(x\right)=-ln\left(x\right)+ln\left(1-x\right)=-ln\left(\frac{x}{1-x}\right)=-logit\left(x\right)[/math][br][br][math]ddH\left(x\right)=\frac{-1}{x\left(1-x\right)}[/math][br][br]which is the core form of [url=https://en.wikipedia.org/wiki/Beta_function]beta function[/url]:[br][br][math]B\left(p,q\right)=\int_0^1x^p\left(1-x\right)^qdx[/math]

Reference: [color=#333333][url=https://math.stackexchange.com/questions/64716/approximating-the-logarithm-of-the-binomial-coefficient/4833062#4833062]Math StackExchange: Approximating the logarithm of the binomial coefficient[/url][/color]