Sum of binomial coefficients with both entries odd

Let $S$ be the sum of $\binom{n}{k}$ over all odd positive integers $n$ and $k$ not exceeding $1000$. Writing $S=\dfrac{a^b-c}{d}$ with integers $a,b,c,d$ and $b$ as large as possible, find $abcd$.

$6000$. For a fixed odd $n$, the sum of $\binom nk$ over odd $k$ is $2^{n-1}$, since half of all subsets have odd size. Summing over $n=1,3,\dots,999$ gives $\sum_{j=0}^{499}4^j=\tfrac{2^{1000}-1}{3}$, so with $b$ maximal $(a,b,c,d)=(2,1000,1,3)$ and $abcd=6000$.