Problem 427

n-sequences

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.

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