An efficient way to numerically compute Stirling numbers of the second kind?

Question

Is there an efficient way to numerically compute Stirling numbers of the second kind?

An approximate (not exact) method would suffice. Something similar to the connection between factorials and gamma functions would work for me.

I googled it a bit and didn't find anything too helpful.

Answer 1

You can use the following recurrence relation

$\mathop{S}\nolimits\left(n,k\right)=k\mathop{S}\nolimits\left(n-1,k\right)+\mathop{S}\nolimits\!\left(n-1,k-1\right)$

see DLMF equation 26.8.E22

and build the (triangular) table.

Answer 2

The accepted answer takes $O(kn)$ multiplications and additions. There is also a method using $O(k\log n)$ additions, multiplications and divisions, using the below identity. $$ S(n,k)=\frac1{k!}\sum_{j=0}^k(-1)^j\binom{k}j(k-j)^n $$

We can use DLMF Equation 26.3.E6 as a recurrence relation to greatly reduce the amount of computation necessary to compute the binomial coefficient.

$$ \binom{m}{n} = \frac{m-n+1}{n}\binom{m}{n-1} $$

Pseudocode:

accum ← 0
k_choose_j ← 1
for j = 0,1,...,k-1:
  accum ← accum + (-1)^j * k_choose_j * (k - j)^n
  k_choose_j ← k_choose_j * (k-j) / (j+1)
return accum / factorial(k)

Note that in the above pseudocode, we update k_choose_j at the end of each iteration, so we need to substitute j + 1 for n in the above relation.