How to determine if $c_0 = 0$ ?
Where
$$F(z) = \left( \prod_{k = 0}^{N} (1+z^{2a_k}) \right) \left( \prod_{j = 0}^{M} (z^{-2b_j} + z^{-2d_j}) \right) = \sum_{n=-Large1}^{Large2} {c_nz^{2n}}$$
and $N,M,a_k,b_j,d_j \in \mathbb{N}$ and are known.
$N$ and $M$ are approximately $1000$ or larger.
$a_k$ and $b_j$ can be very large.
- Expanding both products would result in approximately $2^{N+M}$ terms and is intractable.
- Numerical quadrature around a complex contour around $z = 0$ fails due to high frequency oscillation caused by large $a_k$ ,$b_j$ and $d_j$.
Addendum Subset Sum problem
@mercio identified the subset sum character of the problem.
Let $S = \{a_k\}$ , $a_k \in \mathbb{Z}$ be a set of integers.
Find a non empty subset of elements of $S$ that add to $0$.
A generating function for all the sums, excluding the empty set ($-1$) can be written as:
$$F(z) = -1 + \prod_{a_k \in S} (1+z^{a_k}) = \sum_{n} c_n z^n$$
If $c_0 \ne 0$ then a solution exists.
Each $a_k$ can be left out and $c_0$ redetermined to find a solution.
$$c_n = \frac1{2\pi i} \oint \frac{F(z)}{z^{n+1}} dz$$