Infinite Sequences and Series · Challenge problem

For a positive integer $m$, the generating-function identity $\displaystyle\sum_{n=0}^{\infty}\binom{n+m}{m}x^n=\dfrac1{(1-x)^{m+1}}$ holds for $|x|<1$. Using $m=3$, find the exact value of $\displaystyle\sum_{n=1}^{\infty}\binom{n+3}{3}\left(\dfrac12\right)^n$.