A binomial sum related to Wolstenholme's theorem

A certain alternating sum u(n) of n+1 products of two binomial coefficients has a property similar to Wolstenholme's theorem, namely for all primes p⩾5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in question will be characterized. We also study the sequence u(n) in greater detail, especially its growth and its sign distribution.