## n-sequences

### Problem 427

Published on Saturday, 11th May 2013, 10:00 pm; Solved by 141; Difficulty rating: 95%A sequence of integers S = {s_{i}} is called an `n-sequence` if it has `n` elements and each element s_{i} satisfies 1 ≤ s_{i} ≤ `n`. Thus there are `n`^{n} 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.