counting monotonically increasing sequence that for each object a(i) < i how many monotonically increasing sequence of wholes postive numbers are existing 
that for each object in the sequence 0 $\leq$ a(0) $\leq$ a(1) $\leq$ a(2)....$\leq$ a(n)
and a(i) $<$ i 

my guess its releted to catalan numbers
 A: Hint
The Catalan numbers enumerate the set of lattice paths from $(0,0)$ to $(n,n)$, where each step is one unit up or right, and which stay below the line $y=x$. Try to find a bijection between such paths and the sequences you are counting. The conditions "$a(i)<i$" and "the path must be below the line $y=x$" are strikingly similar$\dots$
Further hint
When $n=2$, there are two possible lattice paths:
   |      _|
_ _|    _|  

and there are two possible sequences:
$$
(0,0) \qquad and\qquad (0,1)
$$
Notice that the path on the left stays pretty low to the ground (until the very end), while the path on the right is gradually increasing. Similarly, the sequence $(0,0)$ stays low, while $(0,1)$ gradually increases. 
When $n=3$, there are $5$ paths:
     |       |     _|       |       _|
     |      _|    |      _ _|     _|
_ _ _|  _ _|   _ _|    _|       _|

sorted in terms of how quickly they increase. Similarly, there are $5$ sequences:
$$
(0,0,0)\quad (0,0,1)\quad(0,0,2)\quad(0,1,1)\quad(0,1,2)
$$
Try to see show the general trend of the paths corresponds to the general trend of the sequences, and use that to construct a bijection that works for any $n$.
