Problem 992

Another Frog Jumping

There are $n+1$ stones in a pond, numbered $0$ to $n$.

A frog starts by jumping onto stone $0$. It then jumps between the stones, only ever jumping to adjacent ones. For fixed $k$, it makes exactly $k+i$ visits to each stone $i$ for $0 \le i \lt n$; however, there are no restrictions on the number of times stone $n$ is visited. The frog can finish on any stone.

If $n=3$ and $k=2$ it would visit stone $0$ two times, stone $1$ three times and stone $2$ four times.

One way of achieving this is:
$0 \to 1 \to 0 \to 1 \to 2 \to 3 \to 2 \to 1 \to 2 \to 3 \to 2$.

Let $J(n,k)$ be the number of ways the frog can make such a journey. For example, $J(3,2) = 17$, $J(6,1) = 1320$ and $J(6,5) = 16793280$.

Find $\displaystyle \sum_{s=0}^4 J(500,10^s)$. Give your answer modulo $987898789$.

又一个青蛙跳跃问题

池塘中有$n+1$块石头，编号为$0$到$n$。

一只青蛙首先跳上石头$0$，然后开始在石头之间跳跃，但每次只能跳到相邻的石头上。对于给定的$k$，它必须恰好跳到石头$i$上$k+i$次，其中$0 \le i \lt n$；但它可以跳到石头$n$上任意次。这段跳跃旅程可以在任意一块石头上结束。

若$n=3$，$k=2$，则石头$0$必须被跳到过两次，石头$1$被跳到过三次，石头$2$被跳到过四次。

其中一种可行的跳跃方案为：
$0 \to 1 \to 0 \to 1 \to 2 \to 3 \to 2 \to 1 \to 2 \to 3 \to 2$。

记$J(n,k)$为青蛙完成这段旅程的可行方案数目。例如，$J(3,2) = 17$，$J(6,1) = 1320$，$J(6,5) = 16793280$。

求$\displaystyle \sum_{s=0}^4 J(500,10^s)$，并对$987898789$取余作为你的答案。