I'm having some trouble finding a closed formula for this quantity:
\begin{align*} \sum_{j} \binom{n}{j}\binom{j}{n-j} \end{align*} I know that this size is equal to \begin{align*} \sum_{j} \binom{n}{j}\binom{n-j}{j} \end{align*} And also \begin{align*} \sum_j \binom{n}{2j}\binom{2j}{j} \end{align*} All these sums are over all allowed $j$, that is, so the lower index in the binomial coefficient is not greater than the upper, and is not negative. Can someone help me? Thanks :)
Edit: It seems like people agree that there might not be a simple closed form for this sum.
Edit: Apparently, this sum is the n-th term in the sequence of central trinomial coefficients, which do not have a closed form that are on my level. I did not know that at the time. Thanks for your answers!