I'm trying to find the number of sequences of integers $0=a_1\leq a_2\leq\ldots\leq a_n$ such that $a_k<k$ for every $1\leq k\leq n$.
I know for $n=3$, the sequences are $ 000, 001, 002, 011,012$, so the number of sequences is $5$, but I don't know how to approach this further.
Any tips?