I was wondering if there exists a nontrivial sequence of numbers $a_{k}$ with the following property:
$$\sum_{k=0}^{N}\binom{n}{k}a_{k}=\binom{2n}{n},$$
where the integer $N=\lfloor\tfrac12n\rfloor$ is $\tfrac12n$ when $n$ is even, and $\tfrac12(n-1)$ when $n$ is odd, and take $n>0$.
By a nontrivial sequence I mean one in which no terms are zero, or if that cannot be done, not all terms but one are zero (the obvious solution).
I guessed that they might be related to Catalan numbers but I'm really not so familiar with their properties and wasn't sure of the best way to approach the question, whether through generating functions or just a clever substitution or something else.
This is only done out of curiosity, so I would consider a reference totally fine, without a proof, but of course understanding why is always great.