Sum of products of all combinations?
Consider the polynomial $p(t) = \prod_{i=1}^n (s_i t + 1)$. Then we want to compute the coefficient of $t^k$ in $p(t)$. We can compute $p(t)$ using fast polynomial multiplication and then output the coefficient of $t^k$. 

Say, partition $n$ polynomials $s_i t + 1$ into pairs and multiply the product of polynomials in each pair (using the standard fast polynomial multiplication algorithm). Then divide the obtained $n/2$ polynomials into pairs, and compute the product in each pair. Repeat this step over and over until we get one polynomial. At each step we can get rid of terms of degree higher than $k$.

Also note that if $k$ is small, we can just use dynamic programming.