Counting Special Non-Descending Sequences

In this blog post, I will discuss the solution to a very fascinating math problem that I came up with while pondering challenging math problems. The solution to this problem happens to be a really interesting combinatorics identity, and it can be used to describe some very interesting counting problems (including the number of ways to match parentheses).
How many non-descending sequences of integers $ a_1, a_2, \dots, a_n $ are there such that $ 0 \leq a_i < i \: \forall \: i \leq n $?

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