G. Partitions
combinatorics; math; number theory
You are given a set of n elements indexed from 1 to n. The weight of i-th element is wi. 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.
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 wi (1 ≤ wi ≤ 109)— weights of elements of the set.
Print one integer — the sum of weights of all partitions of a given set into k non-empty subsets, taken modulo 109 + 7.
Possible partitions in the first sample: {{1, 2, 3}, {4}}, W(R) = 3·(w1 + w2 + w3) + 1·w4 = 24; {{1, 2, 4}, {3}}, W(R) = 26; {{1, 3, 4}, {2}}, W(R) = 24; {{1, 2}, {3, 4}}, W(R) = 2·(w1 + w2) + 2·(w3 + w4) = 20; {{1, 3}, {2, 4}}, W(R) = 20; {{1, 4}, {2, 3}}, W(R) = 20; {{1}, {2, 3, 4}}, W(R) = 26; Possible partitions in the second sample: {{1, 2, 3, 4}, {5}}, W(R) = 45; {{1, 2, 3, 5}, {4}}, W(R) = 48; {{1, 2, 4, 5}, {3}}, W(R) = 51; {{1, 3, 4, 5}, {2}}, W(R) = 54; {{2, 3, 4, 5}, {1}}, W(R) = 57; {{1, 2, 3}, {4, 5}}, W(R) = 36; {{1, 2, 4}, {3, 5}}, W(R) = 37; {{1, 2, 5}, {3, 4}}, W(R) = 38; {{1, 3, 4}, {2, 5}}, W(R) = 38; {{1, 3, 5}, {2, 4}}, W(R) = 39; {{1, 4, 5}, {2, 3}}, W(R) = 40; {{2, 3, 4}, {1, 5}}, W(R) = 39; {{2, 3, 5}, {1, 4}}, W(R) = 40; {{2, 4, 5}, {1, 3}}, W(R) = 41; {{3, 4, 5}, {1, 2}}, W(R) = 42.