Binomial Coefficient Coincidences

These seven equations between binomial coefficients are ‘coincidences’: they aren’t among the four known infinite families. De Weger conjectured that there are no more such coincidences: • Benjamin M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory 63, no. 2 (1997), 373–386. At that time, he and his collaborators checked there were no others involving binomial coefficients less than 1030. Later they checked that there are none involving binomial coefficients less than 1060: • Aart Blokhuis, Andries Brouwer and Benne de Weger, Binomial collisions and near collisions. So, De Weger’s conjecture stands open. The four infinite families, by the way, are these: and the only nontrivial one: the Lind–Singmaster family involving the Fibonacci numbers where : The first three equations in the Lind–Singmaster family are these: But here’s the question I’m interested in: Is there any good explanation for the seven binomial coefficient coincidences?…