Proving Binomial identities using generating functions.

Question

Prove that : $$\binom {2n}{n}=\sum_{r=0}^n \left[\binom nr\right]^2.$$

Help me solving using generating functions.

I tried solving it as follows -

$$\sum_{r=0}^{n}\binom{n}{r}^2=\sum_{r=0}^{n}\binom{n}{r}\binom{n}{n-r}.$$

Now I dont know how to proceed by summing the coefficients to prove it using generating functions.

Answer 1

HINT:

Note that

$$(1+x)^{2n}=(1+x)^{n}(1+x)^n = \left(\sum\limits_{k=0}^{n}\binom{n}{k}x^{k}\right) \cdot \left(\sum\limits_{k=0}^{n}\binom{n}{k}x^{k}\right)$$

From the LHS it is clear that the coefficient before $x^n$ is $\binom{2n}{n}$, and from the RHS it is clear that it is just a sum of products of coefficients before $x^{k}$ and $x^{n-k}$ ($k = 0..n$) and thus it is equal to

$$\sum\limits_{k=0}^n\binom{n}{k}\binom{n}{n-k} = \sum\limits_{k=0}^n\left[\binom{n}{k}\right]^2$$