Is there a closed form for the sum of the cubes of the binomial coefficients?

These numbers are called the Franel Numbers. It's proven in (Petkovšek, M., Wilf, H. and Zeilberger, D. (1996). A=B. Wellesley, MA: A K Peters. p. 160) that there is no closed form for these numbers, in terms of the sum of a fixed number of hypergeometric terms.

However, as @Robert_Israel points out, the expression could possibly be represented by different types of closed form.