Since the Stirling numbers are the coefficients of a polynomial of degree $n$ which is already factored, it can be evaluated at the roots of unity in $O(n)$ O(n\log n)$ multiplications. Then, by Fourier transform, the coefficients can be found in another $O(n\log n)$ multiplications, of roughly $O( n)$ bit numbers. This will find an entire row of the Stirling triangle in time $O(n^2 \log^k n),$ or $O(n \log^k n)$ time per Stirling number. The exponent $k$ is something like $2+\epsilon.$
REMARK The recurrence approach takes $O(n^2)$ arithmetic operations, or $O(n^3)$ bit operations to generate either one, or all of the Stirling numbers, so if the goal is to generate all of them up to a certain size, the simple approach is better. However, if one needs either a single number or a row, the approach I give is considerably faster.