Exercise 2088Combinatorics › Binomial coefficientsM1Added 3 Nov 2026

Generating function of central binomials

#generating function#central binomial#asymptotics

Statement

Show that $\sum_{n\ge0}\binom{2n}{n}x^n=\frac{1}{\sqrt{1-4x}}$ .

Hint

Use the generalized binomial theorem on $(1-4x)^{-1/2}$ .

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Expansion. $(1-4x)^{-1/2}=\sum_{n\ge0}\binom{-1/2}{n}(-4x)^n$ .

Coefficient. $\binom{-1/2}{n}(-4)^n=(-1)^n\binom{2n}{n}4^{-n}(-4)^n=\binom{2n}{n}$ .

Check.
$(-1/2)(-3/2)\cdots(-(2n-1)/2)/n!=(-1)^n\frac{(2n-1)!!}{2^nn!}=\frac{(-1)^n}{4^n}\binom{2n}{n}.$

So $\binom{-1/2}{n}(-4)^n=\binom{2n}{n}$ .

Conclusion. The generating function of central binomials is $(1-4x)^{-1/2}$ .