  We show that for any prime prime $p\not=2$ $$\sum_{k=1}^{p-1} {(-1)^k\over
k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$
by expressing the l.h.s. as a combination of alternating multiple harmonic
sums.
