Sum of square roots of binomial coefficients

Question

Question: let \begin{equation} c_n=\sum_{i=0}^n \sqrt{\binom{n}{i}}. \end{equation} How does $c_n$ grow with $n$?

My conjecture is that $c_n=\Theta(2^{0.5n}n^{0.25})$.This is because \begin{equation} 0.5^{0.5n} c_n = \sum_{i=0}^n \sqrt{\binom{n}{i}0.5^n}=\sum_{i=0}^n\sqrt{Pr(X=i)}, \end{equation} for $X\sim Bin(n,0.5)$. The binomial distribution $Bin(n,0.5)$ is approximately the normal distribution $N(0.5n, 0.25n)$. Also, if $Y\sim N(0.5n, 0.25n)$, it is not hard to see that \begin{equation} \int \sqrt{f(y)} dy =\Theta(n^{0.25}). \end{equation} Therefore I believe that it is also true that $0.5^{0.5n} c_n=\Theta(n^{0.25})$. I played with matlab to get some evidence and found that \begin{equation} \frac{0.5^{0.5n} c_n}{n^{0.25}} \approx \frac{\pi}{2}, \end{equation} for $n=100,1000,10000,100000,1000000$. So the conjecture should be true. So anyone can prove it?

(It turns out that $\frac{0.5^{0.5n} c_n}{n^{0.25}} \approx (2\pi)^{0.25}$ instead of $\frac{\pi}{2}$).

Answer 1

For smallish $k$, we have $$ \binom{n}{n/2+k} \approx \binom{n}{n/2} \exp(-2k^2/n). $$ So $$\sum\sqrt{\binom{n}{n/2+k}} \approx \sqrt{\binom{n}{n/2}} \int_{-\infty}^\infty e^{-k^2/n}\,dk \approx 2^{n/2} (2\pi n)^{1/4}.$$ Of course I did some hand-waving here, but all of this is rigorously justifiable. Use the Euler-Maclaurin theorm to justify replacing the sum by an integral.

Answer 2

Define the complex-variable function $$g(z)=\frac{\sin\pi z}{\pi z}\prod_{k=1}^n\frac1{1-\frac{z}k}.$$ Note $f(k)=\binom{n}k$ for $k\in\mathbb{Z}^{\geq0}$. Moreover, $\sqrt{g(z)}$ is analytic in the region $-1<\Re(z)<n+1$ since $g(z)$ is zero-free there. By the Residue Theorem, $$\sum_{k=0}^n\sqrt{\binom{n}k} =\frac1{2\pi i}\int_C\sqrt{g(z)}\,\frac{\pi\cos\pi z}{\sin\pi z}\,dz$$ where $C$ is a closed (positive) contour inside $-1<\Re(z)<n+1$ going around once about each integer $0\leq k\leq n$. Choose $C_R$ as the rectangular along the lines $\Re(z)=-\frac12$, $\Re(z)=n+\frac12$ and $\Im(z)=\pm R$. Thus $$\int_{C_R}\sqrt{g(z)}\,\frac{\pi\cos\pi z}{\sin\pi z}\,dz =\int_{C_R}\prod_{k=1}^n\sqrt{\frac1{1-\frac{z}k}}\,\frac{\sqrt{\pi}\cos\pi z} {\sqrt{z\sin\pi z}}\,dz.$$ So, letting $R\rightarrow\infty$, we arrive at $$\sum_{k=0}^n\sqrt{\binom{n}k} =\frac1{2\pi i}\left[\int_{-\frac12+i\infty}^{-\frac12-i\infty} +\int_{n+\frac12-i\infty}^{n+\frac12+i\infty} \sqrt{g(z)}\,\frac{\pi\cos\pi z}{\sin\pi z}\,dz\right].$$ The integrand is invariant under $z\rightarrow n-z$, hence it suffices to compute (actually estimate) one of the integrals.

This is where I left off. Perhaps someone can complete the argument.