  We show how to find series expansions for $\pi$ of the form
$\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some
polynomial in $n$ (depending on $m,p,a$). We prove that there exist such
expansions for $m=8k$, $p=4k$, $a=(-4)^k$, for any $k$, and give explicit
examples for such expansions for small values of $m$, $p$ and $a$.
