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$$ \sum_{k>=1}^{\infty} {2N \choose N-k}k $$

How to find this sum?

I know that the answer is $ \frac{1}{2}N{2N \choose N}$

But it is very interesting to know the solution :)

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Our first goal in dealing with this sum is to get rid of the $k$. The standard approach is to rewrite something like $\binom{n}{k} \cdot k$ as $\frac nk \binom{n-1}{k-1} \cdot k$, or $n \binom{n-1}{k-1}$. Here, the bottom index doesn't match the extra factor, but we can make it so with a little extra work: \begin{align} \binom{2N}{N-k} k &= \binom{2N}{N+k}k \\ &= \binom{2N}{N+k}(N+k) - \binom{2N}{N+k} N \\ &= \frac{2N}{N+k} \binom{2N-1}{N+k-1}(N+k) - N \binom{2N}{N+k} \\ &= 2N \binom{2N-1}{N+k-1} - N \binom{2N}{N+k}. \end{align} At this point, we have two sums that are both easier to deal with: $$\sum_{k \ge 1} \binom{2N}{N-k} k = 2N \sum_{k \ge 1} \binom{2N-1}{N+k-1} - N \sum_{k \ge 1} \binom{2N}{N+k}.$$ The sum of all binomial coefficients of the form $\binom{2N-1}{i}$ is $2^{2N-1}$, and our first sum takes only those binomial coefficients of this form where $i \ge N$. These are the second half, which by symmetry is equal to the first half, so the first sum simplifies to $2^{2N-2}$.

We're in much the same position with the second sum, except that the $\binom{2N}{i}$ coefficients also have a central coefficient $\binom{2N}{N}$, which is left out here. The sum of all coefficients that aren't the central one is $2^{2N} - \binom{2N}{N}$, and this sum is half of that.

Putting these facts together, we get \begin{align} \sum_{k \ge 1} \binom{2N}{N-k} k &= 2N \Bigg(2^{2N-2}\Bigg) - N \Bigg(2^{2N-1} - \frac12\binom{2N}{N}\Bigg) \\ &= N \cdot 2^{2N-1} - N \cdot 2^{2N-1} + \frac N2 \cdot \binom{2N}{N} \\ &= \frac N2 \cdot \binom{2N}{N}. \end{align}

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Here is a slightly different variation.

We obtain \begin{align*} \sum_{k=1}^\infty\binom{2N}{N-k}k&=\sum_{k=0}^N\binom{2N}{N-k}k\tag{1}\\ &=\sum_{k=0}^N\binom{2N}{k}(N-k)\tag{2}\\ &=N\sum_{k=0}^N\binom{2N}{k}-\sum_{k=1}^N\binom{2N}{k}k\\ &=N\left(2^{2N-1}+\frac{1}{2}\binom{2N}{N}\right)-2N\sum_{k=1}^N\binom{2N-1}{k-1}\tag{3}\\ &=2^{2N-1}N+\frac{N}{2}\binom{2N}{N}-2N\sum_{k=0}^{N-1}\binom{2N-1}{k}\tag{4}\\ &=2^{2N-1}N+\frac{N}{2}\binom{2N}{N}-2N\cdot2^{2N-2}\tag{5}\\ &=\frac{N}{2}\binom{2N}{N} \end{align*}

Comment:

  • In (1) we set the upper limit of the sum to $N$ since $\binom{2N}{N-k}=0$ for $k>N$. We also start with $k=0$ without changing anything since we are adding zero only.

  • In (2) we change the order of summation $k\rightarrow N-k$.

  • In (3) we use the symmetry of the binomial coefficients and the fact that they sum up to $2^{2N}$. We have additionally to respect the central binomial coefficient since we have an odd number of summands. We also use the binomial identity $\binom{p}{q}=\frac{p}{q}\binom{p-1}{q-1}$ for the right-hand sum.

  • In (4) we shift the index of the sum by $1$.

  • In (5) we again use the symmetry of the binomial coefficients and the fact that they sum up to $2^{2N-1}$. This time we have an even number of summands and need not to care for a central binomial coefficient.

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