Is it true that if $n = 2^k$ for some integer $k\geq 1,$ then $\binom{n}{\frac{n}{2}}$ is an even number?
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
140632
A problem on Binomial Theorem
68400
Proof that $\frac{(2n)!}{2^n}$ is integer
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
104920
Can we find positive integers $a$ and $k \geq 2$ with $2^n - 1 = a^k$?
155150
Division of Factorials [binomal coefficients are integers]
107258
Proving If $k \le \left\lfloor \frac{n}{2} \right\rfloor$ then $\binom{n}{k-1} < \binom{n}{k}$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
123267
Proving that even numbers equal the sum of two odd numbers.
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
139074
Question about $a_n=\sum_{k=1}^n \frac{n}{n^2+k}\ $for $ n \in\mathbb{N}$
118571
Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
96097
Let $k$ be an odd positive integer. Prove that $(1+2+3...+n)|(1^k+2^k+3^k+...+n^k)$ for all positive integers $n$.
131468
Summation of Central Binomial Coefficients divided by even powers of $2$
111951
Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
129680
Prove that $n$ is even.