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!