In how many ways can I write $0$ as a sum of $n\; 0s, 1s \;\text{and}\; -1s?$ In how many ways can I write $0$ as a sum of $n\; 0s, 1s \;\text{or}\; -1s?$  (Taking the order into account).
I suspect there is no closed formula to express the result, but I'd like someone to confirm it, or deny it.
Edit:
e.g., if $n=3$
$$\begin{aligned}0&=\;\;\;0+0+0,\\ 0&=\;\;\;1-1+0,\\ 0&=\;\;\;1+0-1,\\ 0&=\;\;\;0+1-1,\\ 0&=-1+1+0,\\ 0&=-1+0+1,\\ 0&=\;\;\;0-1+1.\end{aligned}$$
So for $n=3\;$ there are $7$ ways.

The number of $0s, 1s$ and $-1s$ possible for a particular $n$ can be seen by the number of solutions to:
$$2p+q=n| p,q \in \Bbb Z^+$$
This can be done in $\frac{n+1}{2}$ ways for odd $n$ and $\frac{n+2}{2}$ ways for even $n$
You'll just need to account for positioning after this.
