Two Arrays
题意翻译
输入两整数 $n$、$m$,求有多少对正整数序列 $a_i$、$b_i$ 满足:
* $a_i$、$b_i$ 的长度为 $m$;
* $a_i$、$b_i$ 中所有元素的值小于等于 $n$;
* 对于所有 $1\le i\le m$,$a_i\le b_i$;
* 序列 $a$ 单调不降,序列 $b$ 单调不升。
答案对 $10^9+7$ 取模。
题目描述
You are given two integers $ n $ and $ m $ . Calculate the number of pairs of arrays $ (a, b) $ such that:
- the length of both arrays is equal to $ m $ ;
- each element of each array is an integer between $ 1 $ and $ n $ (inclusive);
- $ a_i \le b_i $ for any index $ i $ from $ 1 $ to $ m $ ;
- array $ a $ is sorted in non-descending order;
- array $ b $ is sorted in non-ascending order.
As the result can be very large, you should print it modulo $ 10^9+7 $ .
输入输出格式
输入格式
The only line contains two integers $ n $ and $ m $ ( $ 1 \le n \le 1000 $ , $ 1 \le m \le 10 $ ).
输出格式
Print one integer – the number of arrays $ a $ and $ b $ satisfying the conditions described above modulo $ 10^9+7 $ .
输入输出样例
输入样例 #1
2 2
输出样例 #1
5
输入样例 #2
10 1
输出样例 #2
55
输入样例 #3
723 9
输出样例 #3
157557417
说明
In the first test there are $ 5 $ suitable arrays:
- $ a = [1, 1], b = [2, 2] $ ;
- $ a = [1, 2], b = [2, 2] $ ;
- $ a = [2, 2], b = [2, 2] $ ;
- $ a = [1, 1], b = [2, 1] $ ;
- $ a = [1, 1], b = [1, 1] $ .