Problem 427
n-sequences
A sequence of integers S = {si} is called an n-sequence if it has n elements and each element si satisfies 1 ≤ si ≤ n. Thus there are nn distinct n-sequences in total. For example, the sequence S = {1, 5, 5, 10, 7, 7, 7, 2, 3, 7} is a 10-sequence.
For any sequence S, let L(S) be the length of the longest contiguous subsequence of S with the same value. For example, for the given sequence S above, L(S) = 3, because of the three consecutive 7’s.
Let f(n) = ∑ L(S) for all n-sequences S.
For example, f(3) = 45, f(7) = 1403689 and f(11) = 481496895121.
Find f(7 500 000) mod 1 000 000 009.
n-序列
整数序列S = {si}被称为n-序列，如果它有n个元素，而且每个元素si满足1 ≤ si ≤ n。因此，一共有nn个不同的n-序列。例如，序列S = {1, 5, 5, 10, 7, 7, 7, 2, 3, 7}就是一个10-序列。
对于任意序列S，记L(S)是其中相同元素连续出现的最大次数。例如，对于上述序列S，L(S) = 3，因为其中有连续出现的3个7。
对于所有的n-序列S，记f(n) = ∑ L(S)。
例如，f(3) = 45，f(7) = 1403689，f(11) = 481496895121。
求f(7 500 000) mod 1 000 000 009。