How to calculate the sum of binomials?

Question

I want to prove below:

n is natural number.

$$\sum_{k=1}^n k \binom{2n}{n+k} =\frac{1}{2}(n+1) \binom{2n}{n+1}$$

Please tell me above proof.

Answer 1

You can use $$ \sum_{k=0}^{2n}\binom{2n}{k}=(1+1)^{2n}=4^n $$ and $$ \binom{m}{j}=\binom{m}{m-j} $$ and $$ j\binom{m}{j}=m\binom{m-1}{j-1} $$ to simplify your expression.

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For instance, combining them to \begin{align} 2k\binom{2n}{n+k}&=(n+k)\binom{2n}{n+k}-(n-k)\binom{2n}{n-k}\\[1em] &=2n\binom{2n-1}{n+k-1}-2n\binom{2n-1}{n-k-1}\\[1em] &=2n\binom{2n-1}{n+k-1}-2n\binom{2n-1}{n+k} \end{align} Perhaps this can be used in a telescoping sum.