I am trying to count the number of elements in $$ S_{n,k} = \left\{(n_1,\ldots, n_k): \forall i,n_i \in \mathbb Z, n_i \geq 0, \sum_i n_i = n\right\},\\ T_{n,k} = \left\{(n_1,\ldots, n_k): \forall i,n_i \in \mathbb Z, \sum_i |n_i| = n\right\},\\ $$
I have already find out the size of $S_{n,k}$ here. But I find it hard to find a simple expression for $T_{n,k}.$ To express $T_{n,k}$ in terms of $S_{n,k},$ one need to do lots of overcounting and inclusion-exclusion, and apparently it is hard to find a closed form.
Does a nice expression for the size $|T_{n,k}|$ exists?
In this answer, Christian Blatter proved $$|T_{n,k}|=\sum_{j=0}^{\min(n,k)}\binom nj\binom kj2^j.$$ This cannot be written without a summation, which you can prove using Zeilberger's algorithm and Petkovšek's algorithm.
This formula reveals a symmetry between $k$ and $n$ which is not at all obvious from the definition.