Generating functions

See how multiplication with $\frac{1}{x}$ summates the sequence upto n.
$
\begin{matrix}
\text{sequence} & \text{generating function} & \text{formula}\\
1,1,1, \cdots & \frac{1}{1-x} & f(n)=1\\
1,2,3, \cdots & \frac{x}{(1-x)^2} & f(n)=n\\
1,3,6, \cdots & \frac{x}{(1-x)^3} & f(n)=\frac{n(n+1)}{2}\\
1,4,10, \cdots & \frac{x}{(1-x)^4} & f(n)=C(n+3,n)
\end{matrix}
$