combinatorics - Proof of $sum_{k=1}^n binom{n}{k} binom{n}{n+1-k} = binom{2n}{n+1}$ via induction

I'm having trouble to prove the following formula using Induction on $n \in \mathbb{N}$:

$$\sum_{k=1}^n \binom{n}{k} \binom{n}{n+1-k} = \binom{2n}{n+1}.$$
I've tried all the usual identities, but they seem to lead nowhere. Is there any trick to this, or is it just not possible to prove this using induction?

I'm thankful for any tip or advice on how to approach this :)