CF1204E.Natasha, Sasha and the Prefix Sums

Natasha's favourite numbers are $n$ and $1$, and Sasha's favourite numbers are $m$ and $-1$. One day Natasha and Sasha met and wrote down every possible array of length $n+m$ such that some $n$ of its elements are equal to $1$ and another $m$ elements are equal to $-1$. For each such array they counted its maximal prefix sum, probably an empty one which is equal to $0$ (in another words, if every nonempty prefix sum is less to zero, then it is considered equal to zero). Formally, denote as $f(a)$ the maximal prefix sum of an array $a_{1, \ldots ,l}$ of length $l \geq 0$. Then:

$$f(a) = \\max (0, \\smash{\\displaystyle\\max\_{1 \\leq i \\leq l}} \\sum\_{j=1}^{i} a\_j )$$

Now they want to count the sum of maximal prefix sums for each such an array and they are asking you to help. As this sum can be very large, output it modulo $998\: 244\: 853$.

娜塔莎最喜爱的数字是 $n$ 和 $1$，而萨沙最喜爱的数字是 $m$ 和 $-1$。某天，娜塔莎和萨沙相遇了，并写下了所有长度为 $n+m$ 的数组，其中恰好有 $n$ 个元素等于 $1$，其余 $m$ 个元素等于 $-1$。对每个这样的数组，他们计算其最大前缀和（前缀可以为空，此时其和定义为 $0$；换言之，若所有非空前缀和均小于零，则最大前缀和视为 $0$）。形式化地，记 $f(a)$ 为长度为 $l \geq 0$ 的数组 $a_{1, \ldots ,l}$ 的最大前缀和。则：

$$f(a) = \max (0, \smash{\displaystyle\max_{1 \leq i \leq l}} \sum_{j=1}^{i} a_j )$$

现在，他们希望计算所有此类数组的最大前缀和之和，并请你帮忙求解。由于该和可能非常大，请将结果对 $998\,244\,853$ 取模后输出。

The only line contains two integers $n$ and $m$ ($0 \le n,m \le 2\,000$).

唯一的一行包含两个整数 $n$ 和 $m$（$0 \le n,m \le 2\,000$）。

Output the answer to the problem modulo $998\: 244\: 853$.

将问题的答案对 $998\: 244\: 853$ 取模后输出。

In the first example the only possible array is [-1,-1], its maximal prefix sum is equal to $0$.

In the second example the only possible array is [1,1], its maximal prefix sum is equal to $2$.

There are $6$ possible arrays in the third example:

[1,1,-1,-1], f([1,1,-1,-1]) = 2

[1,-1,1,-1], f([1,-1,1,-1]) = 1

[1,-1,-1,1], f([1,-1,-1,1]) = 1

[-1,1,1,-1], f([-1,1,1,-1]) = 1

[-1,1,-1,1], f([-1,1,-1,1]) = 0

[-1,-1,1,1], f([-1,-1,1,1]) = 0

So the answer for the third example is $2+1+1+1+0+0 = 5$.

在第一个例子中，唯一可能的数组是 [-1,-1]，其最大前缀和等于 $0$。

在第二个例子中，唯一可能的数组是 [1,1]，其最大前缀和等于 $2$。

在第三个例子中，共有 $6$ 个可能的数组：

[1,1,-1,-1]，f([1,1,-1,-1]) = 2

[1,-1,1,-1]，f([1,-1,1,-1]) = 1

[1,-1,-1,1]，f([1,-1,-1,1]) = 1

[-1,1,1,-1]，f([-1,1,1,-1]) = 1

[-1,1,-1,1]，f([-1,1,-1,1]) = 0

[-1,-1,1,1]，f([-1,-1,1,1]) = 0

因此，第三个例子的答案为 $2+1+1+1+0+0 = 5$。