Recursion, Catalan Numbers & Generating Functions · Superchallenge problem

A sequence of $14$ terms, each equal to $+1$ or $-1$, contains seven $+1$'s and seven $-1$'s. How many such sequences have every partial sum (the sum of the first $k$ terms, for $k=1,2,\dots,14$) between $0$ and $3$ inclusive?