Question 105
Let $N$ denote the set of nonnegative integers, and $Z$ denote the set of all integers. If a function $f: N \times Z \rightarrow Z$ satisfies the conditions that
$$
i.\ f(0, 0) = 1,\ f(0, 1) = 1,
$$
$$
ii.\ \text{for all}\ k \notin \{0, 1\},\ f(0, k) = 0\ \text{and}
$$
$$
iii.\ \text{for all}\ n \ge 1\ \text{and}\ k,\ f(n, k) = f(n-1, k) + f(n-1, k-2n),
$$
find the value of the sum $\sum_{k=0}^{\binom{2009}{2}} f(2008, k)$.
$$
i.\ f(0, 0) = 1,\ f(0, 1) = 1,
$$
$$
ii.\ \text{for all}\ k \notin \{0, 1\},\ f(0, k) = 0\ \text{and}
$$
$$
iii.\ \text{for all}\ n \ge 1\ \text{and}\ k,\ f(n, k) = f(n-1, k) + f(n-1, k-2n),
$$
find the value of the sum $\sum_{k=0}^{\binom{2009}{2}} f(2008, k)$.