pr.probability - A Local CLT with large variance

For n an even integer, $0 leq i leq$ ${n} choose{j}$, $1 leq j leq n$ let $X_{i,j}$ be a
random variable taking values $frac{n}{2}-j,0,j - frac{n}{2}$ with equal probability. Let $S_{n}$ be the sum of these $2^{n}$
random variables.

My question is, what is the 'correct' local limit theorem for this sum as n goes to infinity?
That is, what is a local limit theorem that is in some sense sharp?

To those who have not heard of the term: A local limit theorem is one that describes probabilities
of the sum being a specific number rather than being in a region of size roughly proportional
to the square root of the variance.

To those who might think this is a little specialized: You're right, of course. I wanted to
post a question with a concrete answer, and this is the simplest one that seems to be 'on the
edge' of where local CLTs hold. The versions you see in textbooks fail for this (those that
I'm familiar with fail in a few places), but 'just barely'. Also, a local CLT does at least hold
here. I'm interested in other borderline cases as well.

To those who might think this is trivial: It is true that a CLT for this sum follows from
standard textbook theorems (e.g. the Lindeberg CLT). It is even true that the martingale
local CLT can be used to get rates here - unfortunately, they seem to be wrong (that is,
quite far from sharp).