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Problem 969

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Kangaroo Hopping

Starting at zero, a kangaroo hops along the real number line in the positive direction. Each successive hop takes the kangaroo forward a uniformly random distance between $0$ and $1$. Let $H(n)$ be the expected number of hops needed for the kangaroo to pass $n$ on the real line.

If we write $\alpha = H(1)$, then for all positive integers $n$, $H(n)$ can be expressed as a polynomial function of $\alpha$ with rational coefficients. For example $H(3)=\alpha^3-2\alpha^2+\frac{1}{2}\alpha$. Define $S(n)$ to be the sum of all integer coefficients in this polynomial form of $H(n)$. Therefore $S(1)=1$ and $S(3)=1+(-2)=-1$.

You are also given $\displaystyle \sum_{n=1}^{10} S(n)=43$.

Find $\displaystyle\sum_{n=1}^{10^{18}} S(n)$. Give your answer modulo $10^9+7$.

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袋鼠跳过$n$

一只袋鼠从原点出发，沿实数轴正方向跳跃，每次跳跃的距离是一个在$0$到$1$之间均匀分布的随机数。记$H(n)$为袋鼠在实数轴上跳跃的总距离超过整数$n$所需的期望跳跃次数。

若记$\alpha = H(1)$，则对于所有正整数$n$，$H(n)$均可表示为$\alpha$的有理系数多项式。例如，$H(3)=\alpha^3-2\alpha^2+\frac{1}{2}\alpha$。定义$S(n)$为多项式$H(n)$中所有整数系数之和，因此$S(1)=1$，$S(3)=1+(-2)=-1$。

已知$\displaystyle \sum_{n=1}^{10} S(n)=43$。

求$\displaystyle\sum_{n=1}^{10^{18}} S(n)$，并对$10^9+7$取余作为你的答案。

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