Count the number of sequences of integers $a_1, a_2, \dots, a_n$ such that $$ a_1 = 0 \quad\text{and}\quad 0 \leq a_{i+1} \leq a_i + 1 \quad\text{for}\quad 1 \leq i < n. $$
At first, I was looking for a connection with Catalan numbers, since this task is from this topic. But I didn't find it. And then I tried to find some recurrence relation for it to make a generating function and I failed at that as well. Help me please