I'm aware of the replies to this question, but I was not able to apply the suggested methods in my case.
I would like to compute the asymptotics of the resulting sum $S$ when computing the number of possibilities to distribute $k$ indistinguishable objects in $m$ distinguishable boxes with size limitation $R$ (see derivation here), which is given by \begin{align} S_{m,R}(k)=\sum_{t}(-1)^t \binom{m}{t}\binom{m+k-t(R+1)-1}{m-1}\,, \end{align} where I would like to take the limit $m\to\infty$, while keeping $R$ and the ratio $k/m$ fixed. From numerics, I can see explicitly that $S_{m,R}(k)$ rapidly approaches a Gaussian distribution centered around $m R/2$, but I would like to find an analytical form of this asymptotic Gaussian distribution.
If I define the quasi-continuous variable $x=k/m$, I expect the asymptotics \begin{align} S_{m,R}(x)\sim \alpha p(m) e^{\beta m-\gamma(x-x_0)^2 m}\quad\text{as}\quad m\to\infty\,, \end{align} where $p(m)$ is some polynomial in $m$ and $x_0=R/2$, while $\alpha$, $\beta$ and $\gamma$ are some real numbers.
