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.