Assuming that the numbers are positive integers,
This is closely related to the Frobenius Coin Problem which says that there is a maximum number $\displaystyle F$ (called the Frobenius number) which is not representable. It is NP-Hard to find out the Frobenius number when there are at least $\displaystyle 3$ numbers.
For a formula like approach to determine if such a representation is possible or not, you can use generating functions, which can be used to give a pseudo polynomial time algorithm, polynomial in size $\displaystyle W = n_1 + n_2 + \cdots + n_k$.
If the numbers are $\displaystyle n_1, n_2, \dots, n_k$ and you need to see if they can be summed to $\displaystyle S$ then the number of ways it can be done is the coefficient of $\displaystyle x^S$ in
$$\displaystyle (1+x^{n_1} + x^{2n_1} + x^{3n_1} + \cdots )(1+ x^{n_2} + x^{2n_2} + x^{3n_2} + \cdots ) \cdots (1 + x^{n_k} + x^{2n_k} + x^{3n_k} + \cdots )$$
$$\displaystyle = \dfrac{1}{(1-x^{n_1})(1-x^{n_2}) \cdots (1-x^{n_k})}$$
Using partial fractions this can be written as
$$\displaystyle \sum_{j=1}^{m} \dfrac{C_j}{c_j - x}$$
where $\displaystyle C_j$ and $\displaystyle c_j$ are appropriate complex numbers and $\displaystyle m \le n_1 + n_2 + \cdots + n_k$.
The coefficient of $\displaystyle x^S$ is thus given by
$$\displaystyle \sum_{j=1}^{m} \dfrac{C_j}{c_j^{S+1}}$$
which you need to check is zero or not.
Of course, this might require quite precise floating point operations and does not actually tell you what numbers to choose.