Finding a closed formula for a summation with two binomial coefficients Given non-negative integers $m$ and $n$ with $m\leq n$, what is the following equal to?
$$
\sum_{k=0}^mC_m^kC_{n+k}^m={?}\tag{?}
$$
Here, 
$$
C_m^k=\frac{m!}{k!(m-k)!},
$$
so using standard notation for binomial coefficients the summation is
$$
\sum_{k = 0}^{m}{m \choose k}{n + k \choose m}.
$$
As a concrete example, for $(m,n)=(1,2)$, $(1,1)$ and $(2,2)$ one respectively gets following calculations: 
$$
\sum_{k=0}^1C_1^kC_{2+k}^1=2+3=5,\quad  \sum_{k=0}^1C_1^kC_{1+k}^1=1+2=3, \quad \sum_{k=0}^2C_2^kC_{3+k}^2=3+12+10=25.
$$
This problem is whether there is a closed formula for (?).
 A: I don't believe that there is a closed form for this sum, but we can get a generating function as follows:
$$
\begin{align}
\sum_{k=0}^m\binom{m}{k}\binom{n+k}{m}
&=\sum_{k=0}^m\binom{m}{k}\binom{n+k}{n-m+k}\\
&=\sum_{k=0}^m(-1)^{n-m+k}\binom{m}{m-k}\binom{-m-1}{n-m+k}\\
&=\sum_{k=0}^m(-1)^{n-k}\binom{m}{k}\binom{-m-1}{n-k}\\
\end{align}
$$
This is the coefficient of $x^n$ in $(1+x)^m(1-x)^{-m-1}$.

Examples:
$$
\begin{align}
m=1:&&(1+x)^1(1-x)^{-2}&=1+\color{#C00000}{3}x+\color{#C00000}{5}x^2+7x^3+9x^4+11x^5+\dots\\
m=2:&&(1+x)^2(1-x)^{-3}&=1+5x+13x^2+\color{#C00000}{25}x^3+41x^4+\dots
\end{align}
$$
A: Note that ${k\choose m}$ is zero unless $k=m$.  Then your sum simplifies to the single term ${m\choose n+m}$, which is again zero unless $m=n=0$, in which case it's 1.
A: There is no closed formula for this summation. In Concrete Mathematics there is a table (on page 169) of summations involving the products of two binomial coefficients with closed forms; all are variations of the Vandermonde identity. The easy way to remember which ones have closed forms, is that if the summation involves both upper or both lower indices (traversing them in opposite directions in case of upper indices), then without anything else a closed form exists, while if the summation involves mixed positions a closed form exists if (and only if) an alternating sign is present. In your expression there is summation over a lower and an upper index but no alternating sign; there is no closed formula. 
