Today, during the evening chat, Emmanuel Abbe threw an interesting question: Whether the sum of square roots of consecutive binomial coefficients converge to some closed form! That is, ${S(n)=\displaystyle \sum_{k=0}^{n}{\sqrt{\binom{n}{k}}}}$. We tried a few known combinatorics tweak, but no meaningful solution arrived. We were also wondering whether this has some asymptotic limit, but that too did not yield anything. A quick check on Mathematica wasn’t helpful either. Now the question is: Does this sum yield some closed form expression.

While playing with this sum in Mathematica, I found that for the sum of squares of binomial coefficients, there is a nice simple closed form.

${S_{2}(n)=\displaystyle \sum_{k=0}^{n}{{\binom{n}{k}}^{2}}=\binom{2n}{n}}$

I was toying with a proof. It turns out that, the proof is extremely simple and is a one line tweak of the Vandermonde identity ${\binom{p+q}{m}=\displaystyle \sum_{i=0}^{m}{\binom{p}{i}\binom{q}{m-i}}}$. Simply substitute ${p=q=m=n}$ and we have the results on table. The natural question then would be: Is there a generalization for ${ S_{r}(n)=\displaystyle \sum_{k=0}^{n}{{\binom{n}{k}}^{r}}}$ for any ${r\in \mathbb{N}_{\ge 1}}$. Ofcourse now for ${r=1,2}$ it is trivial.

Apparently, it turns out that, there is no closed form expression for a general (all) ${r}$. There are some interesting divisibility properties of these sums. An interesting account of that is addressed by Neil Calkin (Factors of sums of powers of binomial coefficients).

At the moment, I get a feeling that sum of fractional powers of binomial coefficients is not trivial. May be there is no closed form. May be not!