<s>[INST] Binomial series $\sum^{\infty}_{n=0} {{n+k}\choose{k}} a^{n}$ Does someone know how to derive this sum?
$$\sum^{\infty}_{n=0} {{n+k}\choose{k}} a^{n}$$ where $|a|<1$ and $k$ is given integer.
 [/INST] $\binom{n+k}{k} = [x^k]:(1+x)^{n+k}$
\begin{eqnarray*} 
\sum_{n=0}^{\infty} \binom{n+k}{k} a^k &=& [x^k]: \sum_{n=0}^{\infty}  a^k (1+x)^{n+k} \\
&=& [x^k]:    (1+x)^{k} \frac{1}{1-a(1+x)} \\
&=& [x^k]:    (1+x)^{k} \frac{1}{1-a} \frac{1}{1-\frac{ax}{1-a}}\\
&=& \frac{1}{1-a}[x^k]:   \left( \sum_{j=0}^{k} \binom{k}{j} x^{k-j} \right)\left( \sum_{l=0}^{\infty} \left(\frac{ax}{1-a} \right)^l \right) \\
&=&\frac{1}{1-a}\sum_{j=0}^{k} \binom{k}{j} \left(\frac{a}{1-a} \right)^j \\
&=&\frac{1}{1-a}\left(1+ \frac{a}{1-a}\right)^k = \color{blue}{\frac{1}{(1-a)^{k+1}}}
\end{eqnarray*}
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