# modular arithmetic – congruence of Stirling numbers \$ S (n, frac {p-1} {2}) pmod {p} \$

For $$p$$ bonus, is there a simple extension for the Stirling number of the second type $$S (n, k)$$, of shape $$S (n, frac {p-1} {2}) pmod {p}$$? I've tried using the explicit extension $$binom {n} { frac {p-1} {2}} S (n, k) = (-1) ^ { frac {p-1} {2}} sum_ {l = 0} ^ { frac {p-1} {2}} (- 1) ^ l binom {n} {l} l ^ n$$ and a search for literature, but it ended in nothing.