<s>[INST] Closed-form expression of a series involving combinatorics How to find the closed-form expression of the function $$F(x,y)=\sum_{n=0}^\infty\sum_{m=0}^\infty {{m+n}\choose {n}}x^ny^m=1+x+y+x^2+y^2+2xy+...?$$
 [/INST] The inner sum is
$$\sum_{m=0}^\infty \binom{m+n}{n} y^m=\frac{1}{(1-y)^{n+1}}.$$
So
\begin{align}
F(x,y)&=\sum_{n=0}^\infty \frac{1}{(1-y)^{n+1}} x^n \\
&= \frac{1}{1-y}\sum_{n=0}^\infty \left(\frac{x}{1-y}\right)^n \\
&= \frac{1}{1-y}\cdot\frac{1}{1-x/(1-y)} \\
&= \frac{1}{1-x-y}.
\end{align}
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