TITLE: What is $\sum_{i=0}^{n}\binom{n}{i}^3$?
QUESTION [9 upvotes]: We know that $$\sum_{i=0}^{n}\binom{n}{i}=2^n$$
and that
$$\sum_{i=0}^{n}\binom{n}{i}^2= \binom{2n}{n}$$
what about
$$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?

REPLY [4 votes]: These are called the Franel Numbers
As Michael noted, see oeis.org/A000172 for a summary of known facts and references...