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Problem 830

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Binomials and Powers

Let $\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$.

You are given, $S(10)=142469423360$.

Find $S(10^{18})$. Submit your answer modulo $83^3 89^3 97^3$.

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二项式和幂

记$\displaystyle S(n)=\sum\limits_{k=0}^{n}\binom{n}{k}k^n$。

已知$S(10)=142469423360$。

求$S(10^{18})$，并将你的答案对$83^3 89^3 97^3$取余。

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