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Chinese · Translation
Formal · Original
You are given a set of _n_ elements indexed from 1 to _n_. The weight of _i_\-th element is _w__i_. The weight of some subset of a given set is denoted as . The weight of some partition _R_ of a given set into _k_ subsets is (recall that a partition of a given set is a set of its subsets such that every element of the given set belongs to exactly one subset in partition).
Calculate the sum of weights of all partitions of a given set into exactly _k_ **non-empty** subsets, and print it modulo 109 + 7. Two partitions are considered different iff there exist two elements _x_ and _y_ such that they belong to the same set in one of the partitions, and to different sets in another partition.
## Input
The first line contains two integers _n_ and _k_ (1 ≤ _k_ ≤ _n_ ≤ 2·105) — the number of elements and the number of subsets in each partition, respectively.
The second line contains _n_ integers _w__i_ (1 ≤ _w__i_ ≤ 109)— weights of elements of the set.
## Output
Print one integer — the sum of weights of all partitions of a given set into _k_ **non-empty** subsets, taken modulo 109 + 7.
[samples]
## Note
Possible partitions in the first sample:
1. {{1, 2, 3}, {4}}, _W_(_R_) = 3·(_w_1 + _w_2 + _w_3) + 1·_w_4 = 24;
2. {{1, 2, 4}, {3}}, _W_(_R_) = 26;
3. {{1, 3, 4}, {2}}, _W_(_R_) = 24;
4. {{1, 2}, {3, 4}}, _W_(_R_) = 2·(_w_1 + _w_2) + 2·(_w_3 + _w_4) = 20;
5. {{1, 3}, {2, 4}}, _W_(_R_) = 20;
6. {{1, 4}, {2, 3}}, _W_(_R_) = 20;
7. {{1}, {2, 3, 4}}, _W_(_R_) = 26;
Possible partitions in the second sample:
1. {{1, 2, 3, 4}, {5}}, _W_(_R_) = 45;
2. {{1, 2, 3, 5}, {4}}, _W_(_R_) = 48;
3. {{1, 2, 4, 5}, {3}}, _W_(_R_) = 51;
4. {{1, 3, 4, 5}, {2}}, _W_(_R_) = 54;
5. {{2, 3, 4, 5}, {1}}, _W_(_R_) = 57;
6. {{1, 2, 3}, {4, 5}}, _W_(_R_) = 36;
7. {{1, 2, 4}, {3, 5}}, _W_(_R_) = 37;
8. {{1, 2, 5}, {3, 4}}, _W_(_R_) = 38;
9. {{1, 3, 4}, {2, 5}}, _W_(_R_) = 38;
10. {{1, 3, 5}, {2, 4}}, _W_(_R_) = 39;
11. {{1, 4, 5}, {2, 3}}, _W_(_R_) = 40;
12. {{2, 3, 4}, {1, 5}}, _W_(_R_) = 39;
13. {{2, 3, 5}, {1, 4}}, _W_(_R_) = 40;
14. {{2, 4, 5}, {1, 3}}, _W_(_R_) = 41;
15. {{3, 4, 5}, {1, 2}}, _W_(_R_) = 42.
你被给定一个包含 #cf_span[n] 个元素的集合，元素编号从 #cf_span[1] 到 #cf_span[n]。第 #cf_span[i] 个元素的权重为 #cf_span[wi]。一个子集的权重定义为该子集中所有元素权重之和。一个将给定集合划分为 #cf_span[k] 个子集的划分 #cf_span[R] 的权重定义为 （回忆：一个划分是原集合的若干子集的集合，满足原集合中的每个元素恰好属于其中一个子集）。
计算将给定集合划分为恰好 #cf_span[k] 个 *非空* 子集的所有划分的权重之和，并对 #cf_span[109 + 7] 取模输出。两个划分被认为是不同的，当且仅当存在两个元素 #cf_span[x] 和 #cf_span[y]，使得它们在其中一个划分中属于同一个子集，而在另一个划分中属于不同的子集。
第一行包含两个整数 #cf_span[n] 和 #cf_span[k]（#cf_span[1 ≤ k ≤ n ≤ 2·105]）——元素个数和每个划分中的子集个数。
第二行包含 #cf_span[n] 个整数 #cf_span[wi]（#cf_span[1 ≤ wi ≤ 109]）——集合中各元素的权重。
输出一个整数——将给定集合划分为 #cf_span[k] 个 *非空* 子集的所有划分的权重之和，对 #cf_span[109 + 7] 取模。
## Input
第一行包含两个整数 #cf_span[n] 和 #cf_span[k]（#cf_span[1 ≤ k ≤ n ≤ 2·105]）——元素个数和每个划分中的子集个数。第二行包含 #cf_span[n] 个整数 #cf_span[wi]（#cf_span[1 ≤ wi ≤ 109]）——集合中各元素的权重。
## Output
输出一个整数——将给定集合划分为 #cf_span[k] 个 *非空* 子集的所有划分的权重之和，对 #cf_span[109 + 7] 取模。
[samples]
## Note
第一个样例的可能划分： #cf_span[{{1, 2, 3}, {4}}], #cf_span[W(R) = 3·(w1 + w2 + w3) + 1·w4 = 24]; #cf_span[{{1, 2, 4}, {3}}], #cf_span[W(R) = 26]; #cf_span[{{1, 3, 4}, {2}}], #cf_span[W(R) = 24]; #cf_span[{{1, 2}, {3, 4}}], #cf_span[W(R) = 2·(w1 + w2) + 2·(w3 + w4) = 20]; #cf_span[{{1, 3}, {2, 4}}], #cf_span[W(R) = 20]; #cf_span[{{1, 4}, {2, 3}}], #cf_span[W(R) = 20]; #cf_span[{{1}, {2, 3, 4}}], #cf_span[W(R) = 26]; 第二个样例的可能划分： #cf_span[{{1, 2, 3, 4}, {5}}], #cf_span[W(R) = 45]; #cf_span[{{1, 2, 3, 5}, {4}}], #cf_span[W(R) = 48]; #cf_span[{{1, 2, 4, 5}, {3}}], #cf_span[W(R) = 51]; #cf_span[{{1, 3, 4, 5}, {2}}], #cf_span[W(R) = 54]; #cf_span[{{2, 3, 4, 5}, {1}}], #cf_span[W(R) = 57]; #cf_span[{{1, 2, 3}, {4, 5}}], #cf_span[W(R) = 36]; #cf_span[{{1, 2, 4}, {3, 5}}], #cf_span[W(R) = 37]; #cf_span[{{1, 2, 5}, {3, 4}}], #cf_span[W(R) = 38]; #cf_span[{{1, 3, 4}, {2, 5}}], #cf_span[W(R) = 38]; #cf_span[{{1, 3, 5}, {2, 4}}], #cf_span[W(R) = 39]; #cf_span[{{1, 4, 5}, {2, 3}}], #cf_span[W(R) = 40]; #cf_span[{{2, 3, 4}, {1, 5}}], #cf_span[W(R) = 39]; #cf_span[{{2, 3, 5}, {1, 4}}], #cf_span[W(R) = 40]; #cf_span[{{2, 4, 5}, {1, 3}}], #cf_span[W(R) = 41]; #cf_span[{{3, 4, 5}, {1, 2}}], #cf_span[W(R) = 42].
**Definitions**
Let $ n, k \in \mathbb{Z}^+ $ with $ 1 \leq k \leq n \leq 2 \cdot 10^5 $.
Let $ W = (w_1, w_2, \dots, w_n) $ be a sequence of positive integers representing the weights of $ n $ elements.
Let $ \mathcal{P}_{n,k} $ denote the set of all partitions of the set $ [n] = \{1, 2, \dots, n\} $ into exactly $ k $ non-empty, unlabeled subsets.
For a partition $ R \in \mathcal{P}_{n,k} $, define its weight as:
$$
\text{weight}(R) = \sum_{S \in R} \left( \sum_{i \in S} w_i \right)
$$
**Constraints**
1. $ 1 \leq k \leq n \leq 2 \cdot 10^5 $
2. $ 1 \leq w_i \leq 10^9 $ for all $ i \in [n] $
**Objective**
Compute:
$$
\sum_{R \in \mathcal{P}_{n,k}} \text{weight}(R) \mod (10^9 + 7)
$$