A binomial sum

Evaluate the sum

$\sum_{k=0}^n \binom{n}{k}\frac{k!}{(n+1+k)!}$

Solution

We have successively

$\begin{align*} \sum_{k=0}^n \binom{n}{k}\frac{k!}{(n+1+k)!} &= \sum_{k=0}^{n} \frac{n!}{\left ( n-k \right )! \left ( n+k+1 \right )!} \\ &=\frac{n!}{\left ( 2n+1 \right )!} \sum_{k=0}^{n} \binom{2n+1}{n-k} \\ &= \frac{n!}{\left ( 2n+1 \right )!} \sum_{k=0}^{n} \binom{2n+1}{k} \\ &=\frac{n!}{2 \cdot \left ( 2n+1 \right )!} \sum_{k=0}^{2n+1} \binom{2n+1}{k} \\ &= \frac{n! \cdot 2^{2n}}{\left ( 2n+1 \right )!} \end{align*}$

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A binomial sum

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