Counting Solutions to $x_1 + x_2 + \dots + x_k = n$ with $x_i \leq r$ Closed Form A previous question asked how we can calculate the number of positive integer solutions to $x_1 + x_2 + \dots + x_k = n$ where $x_i \leq r.$ The aforementioned question thread gave an answer as
$$\sum^{k}_{t = 0}(-1)^t{{k}\choose{t}}{{n - t(r + 1) + k - 1}\choose{k - 1}}$$
but I was wondering if anybody knew how to calculate a closed form of this sum. I had some difficulty using techniques such as Snake Oil because of the $t(r + 1)$ term in the second binomial coefficient so I wanted to see if anybody else has tried to do this.
 A: Unfortunately, no, it does not have a simpler form. I have been dealing with it for a long time and didn't find anything better in the specialized literature. 
However, for large $n$ it converges quickly to a Gaussian.
In any case I suggest (as in the answer to the other post you cited) to write it in this
alternative way:
$$
N_b (s,r,m)\quad \left| {\;0 \leqslant \text{integers  }s,m,r} \right.\quad  =
\sum\limits_{\left( {0\, \leqslant } \right)\,\,k\,\,\left( { \leqslant \,\frac{s}{r+1}\, \leqslant \,m} \right)} 
{\left( { - 1} \right)^k \binom{m}{k}
 \binom
 { s + m - 1 - k\left( {r + 1} \right) } 
 { s - k\left( {r + 1} \right)}\ }
$$
(where $s$ is your $n$ and $m$ your $k$)
because:
 -  in the way you wrote it, with the upper bound to $m$ you  get false results
(actually you get $0$ because of taking $\Delta ^m$ of a polynomial of degree $m-1$) ;
 -  the upper bound shall be $s/(r+1)$, which is less than $m$;
 - the form suggested above allows to omit the bounds, being implicit in the binomials, and thereby easing the algebraic manipulation.
--  addendum --
I take the chance of your comment to briefly summarize some features of the formula above.
There are alternative formulations, but in fact not simpler, such as
$$
\eqalign{
  & N_b (s,r,m) =   \cr 
  &  = \sum\limits_k {\left( { - 1} \right)^{\;\left\lfloor {{k \over {r + 1}}} \right\rfloor } \;
 \left( \matrix{  m - 1 \cr   \left\lfloor {{k \over {r + 1}}} \right\rfloor  \cr}  \right)
 \left( \matrix{  s + m - 2 - k \cr   s - k \cr}  \right)}  =   \cr 
  &  = \sum\limits_{j,\,k} {\left( { - 1} \right)^{j + k}
 \left( \matrix{  m  \cr   j  \cr}  \right)\left( \matrix{  j(r + 1) \cr   k \cr}  \right)
 \left( \matrix{  s - k + m - 1 \cr   s \cr}  \right)}  =   \cr 
  &  = \left( {1 - E_{\,s} ^{\, - (r + 1)} } \right)^{\,m}
 \left( \matrix{  s + m - 1 \cr   s \cr}  \right)\quad 
 \left| {\;E_{\,s} f(s,m) = f(s + 1,m)} \right. \cr} 
$$
The ogf in $s$ is instead quite simple (re. to this related post)
$$
F_b (x,r,m) = \sum\limits_{0\,\, \leqslant \,\,s\,\,\left( { \leqslant \,\,r\,m} \right)} {N_b (s,r,m)\;x^{\,s} }
  = \left( {1 + x +  \cdots  + x^{\,r} } \right)^m  = \left( {\frac{{1 - x^{\,r + 1} }}{{1 - x}}} \right)^m 
$$
and the medium terms easily show another way to express $N_b$ as a multinomial expansion.
Multiple o.g.f. in $s,m$ follows easily.
Also $N_b$ satisfies some simple relations and recurrences, such as
$$
\eqalign{
  & N_b (s,r,m) = N_b (mr - s,r,m)  \cr 
  & N_b (s,r,m + n) = \sum\limits_l {N_b (l,r,m)\;N_b (s - l,r,n)} \quad  \Leftrightarrow   \cr 
  &  \Leftrightarrow \quad N_{\,b} (s,r,m) - N_{\,b} (s - 1,r,m) = N_{\,b} (s,r,m - 1) - N_b (s - r - 1,r,m - 1)  \cr 
  & N_{\,b} (s,r,m) = \sum\limits_{j,\;k}
  {\left( \matrix{  m \cr   j \cr}  \right)\;N_{\,b} (k,t,m - j)\,N_{\,b} (\,s - k - j(t + 1),r - t - 1,j)\,}
  \quad \left| {\;0 \le t \le r - 1} \right.  \cr 
  & N_{\,b} (s_\, ,r,m) = \left[ {0 = r} \right]\left[ {0 = s} \right] + \sum\limits_k {
 \left( \matrix{  m \cr  k \cr}  \right)N_{\,b} (s - kr_\, ,r - 1,m - k)}  \cr} 
$$
where $[P]$ denotes the Iverson bracket.
Concerning the asymptotics you may refer to this post where it is explained how we reach to
$$
p(s,r,m) = {{N_{\,b} (s,r,m)} \over {\left( {r + 1} \right)^{\,m} }}\;\; \to \;{\cal N}\left( {m{r \over 2},\;m{{\left( {r + 1} \right)^{\,2}  - 1} \over {12}}} \right)
$$
approximating the sum of $m$ i.i.d discrete uniform variables on $[0,r]$, as the sum of 
$m$ continuous uniform variables (the Irwin-Hall distribution) and then of $m$ normal variables with same mean and variance.
As for the literature finally, for the $N_b$ itself there is not much more than the Mathpages article.
However $N_b$ is a building block for some related problems arising in a number of applications.
That comes mainly from another interpretation of $N_b(s,r,m+1)$ as the
Number of  binary strings,  with $s$ "$1$"'s and $m$ "$0$"'s in total, that have up to $r$ consecutive $1$s
as explained in this post.
So there is nowadays quite a vast literature dealing with it from different perspectives in the fields of:
 - digital transmission - error bursts (which was the origin of my interest in it some decades ago);
 - system reliability, the so called consecutive k-out-of-n:F systems;
 - stochastic processes, queue theory;
 - the so called k-order extension of some common distributions;
 - it is intimately related to n-bonacci numbers;
 - quality control, consecutiveness of defects in a sequential batch;
...  etc. 
Most of my links have become obsolete, but searching on the above subjects starting from the few links I gave
you can find various papers on the aspects of best interest to you.
