Closed form or limiting form of an expression involving binomial coefficients

2018-03-12 20:22:04

This question leads to an application of the inclusion‒exclusion principle leading to this sum:

$$

\sum_{k=0}^n (-1)^k \binom n k (n-k)^x = (-1)^n \sum_{k=0}^n (-1)^k \binom n k k^x

$$

$$

\text{e.g. } \quad 1\cdot 6^{10} - 6\cdot 5^{10} +15\cdot 4^{10} - 6\cdot3^{10} + 1\cdot2^{10}

$$

Is there either a closed form or some sort of limiting form as $n\to\infty\text{?}$

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    $\ds{\sum_{k = 0}^{n}\pars{-1}^{k}{n \choose k}\pars{n - k}^{x} =

    \pars{-1}^{n}\sum_{k=0}^{n}\pars{-1}^{k}{n \choose k}k^{x}:\ {\Large ?}}$.

    \begin{align}

    &\bbox[10px,#ffd]{\ds{%

    \pars{-1}^{n}\sum_{k = 0}^{n}\pars{-1}^{k}{n \choose k}k^{x}}} =

    \pars{-1}^{n}\sum_{k = 0}^{n}\pars{

    2018-03-12 22:26:36