<s>[INST] Is it possible to find a closed-form expression for this summation? Is it possible to find a closed-form expression for this summation?
$$\sum\limits_{k = 0}^n {\left( \left( {\matrix{
   2n  \cr 
   k  \cr 
 } } \right)+ \left( {\matrix{
   2n  \cr 
   n+k-1  \cr 
 } } \right) \left( \frac{1}{3}\right)^\left(2n-2k+1 \right) \right) 
\cdot\left( \frac{1}{3}\right)^k}$$
I can not find one that is not using hyper geometric functions, and splitting the sum did not help me, so there might be something I am missing. If anyone has advice that would be great.
 [/INST] When you have this kind of summation, you cannot avoid at least gaussian hypergeometric functions and in most cases you will not obtain any closed form.
Relacing the $\frac 13$ by $x$ and splitting the sum, you have
$$\sum_{k=0}^n  \binom{2 n}{k} x^k=(x+1)^{2 n}-\binom{2 n}{n+1} x^{n+1} \, _2F_1(1,1-n;n+2;-x)$$ The second term will be
$$\sum_{k=0}^n \binom{2 n}{n+1-k} x^{2 n-k}=\binom{2 n}{n+1} x^{2 n} \, _2F_1\left(1,-n-1;n;-\frac{1}{x}\right)-x^{n-1}$$ probably leading to what Wolfram Alpha gave for your specific case.
Now, for a given value of $n$, what you will obtain is a polynomial of degree $2n$ in $x$ with integer coefficients. The very first ones are
$$\left(
\begin{array}{cc}
 n & \text{Polynomial} \\
 1 & x^2+4 x+1 \\
 2 & 4 x^4+6 x^3+10 x^2+4 x+1 \\
 3 & 15 x^6+20 x^5+15 x^4+26 x^3+15 x^2+6 x+1 \\
 4 & 56 x^8+70 x^7+56 x^6+28 x^5+78 x^4+56 x^3+28 x^2+8 x+1 \\
 5 & 210 x^{10}+252 x^9+210 x^8+120 x^7+45 x^6+262 x^5+210 x^4+120 x^3+45 x^2+10
   x+1 
\end{array}
\right)$$
Very few patterns have been found in $OEIS$.
</s>