TR-MO 2008 Problem 4

$f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions

$a)$ $f(0,0)=1$ , $f(0,1)=1$ ,

$b)$ $\forall k \notin \left\{0,1\right\}$ $f(0,k)=0$ and

$c)$ $\forall n \geq 1$ and $k$ , $f(n,k)=f(n-1,k)+f(n-1,k-2n)$

find the sum $\sum_{k=0}^{\binom{2009}{2}}f(2008,k)$