Convolution of binomial coefficients As part of a (SE) problem I've been working on, I came up with this expression:
$$
\sum_{i=0}^M\binom{M-1+i}{i}\binom{M+i}{i}
$$
I'd like to get a closed form for this, but after a considerable amount of time searching my references and online sources (not to mention the time I've spent bashing this into other equally opaque equivalences), I've come up empty. Does anyone have a clue? I'll be happy to link to the original if asked, but the expression more or less tells the story.

If the “$-1$” were missing from the first binomial, we'd get OEIS A$112029$, which converges 
asymptotically to $\dfrac{4^{2n+1}}{3\pi~n}~,$ and –at the same time– we also have $\displaystyle\lim_{n\to\infty}\frac{A_{112029}(n)}{a_n}=2$, where 
$a_n$ is our sequence. So, in conclusion, $a_n\sim\dfrac{2^{4n+1}}{3\pi~n}$ . It does not appear to possess a meaningful 
closed form expression, and even the simpler afore-mentioned OEIS sequence seems to lack one.
