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Problem 830

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Binomials and Powers

Let ${\displaystyle S(n)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})k^n}$.

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${\displaystyle S(n)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})k^n}$。

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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