TITLE: Closed form for $\prod_{k=0}^n\binom{n}{k}x^ky^{n-k}$.
QUESTION [5 upvotes]: This is well known that we have :
$$\forall (x,y) \in \mathbb{C}^2, ~ \forall n \in \mathbb{N}^*, ~ (x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}$$

Now I don't know if there is any possible application but in same conditions (at least for real numbers) is there a close form to this one? 
$$\prod_{k=0}^n\binom{n}{k}x^ky^{n-k}$$

For real numbers I think in all cases (but I am not sure) if there is a close the form may look like $A\times x^a y^b$ where $A$ is an integer from the product of the binomial terms.
I looked at some values on Wolfram-Alpha with $(x,1-x)$ and $|x|<1$ for fun but I have some difficulties to find a close form a least to $A$. 
I would be gald if someone has an idea.

REPLY [6 votes]: To complement Masacroso answer, note that the product of $k^k$ can be rewritten as
$$
\eqalign{
  & \prod\limits_{k = 1}^n {k^{\,k} }  = \prod\limits_{k = 1}^n {\left( {{{\Gamma \left( {k + 1} \right)} \over {\Gamma \left( k \right)}}} \right)^{\,k} }  =   \cr 
  &  = {{\Gamma (2)^{\,1} } \over {\Gamma (1)^{\,1} }}{{\Gamma (3)^{\,2} } \over {\Gamma (2)^{\,2} }}{{\Gamma (4)^{\,3} } \over {\Gamma (3)^{\,3} }}\; \cdots \;{{\Gamma (n + 1)^{\,n} } \over {\Gamma (n)^{\,n} }} = \;  \cr 
  &  = {{\Gamma (n + 1)^{\,n} } \over {\prod\limits_{1\, \le \,k\, \le \;n} {\Gamma (k)} }} = {{G(1)} \over {G(n + 1)}}\left( {n!} \right)^{\,n}  \cr} 
$$
where $G(x)$ denotes the Barnes G_Function