Eccoci.

A sequence of integers $S = \{s_i\}$ is called an $n$-sequence if it has $n$ elements and each element $s_i$ satisfies $1 \leq s_i \leq n$. 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 $f(n) = \sum L(S)$ for all $n$-sequences S.

For example, $f(3) = 45$, $f(7) = 1403689$ and $f(11) = 481496895121$.

Find $f(7\,500\,000) \bmod 1\,000\,000\,009$.