Divisibility and Binomial Coefficients

[Stefan Steinerberger]

First, thanks for all your interest and thoughts.
The computed numbers lead me to conjecture that
both problems

C(n,a)+C(n,a+1) | C(n,2a+1)
C(n,a)+C(n,a+2) | C(n,2a+2)

have infinitely many solutions and that Drew's observation
about the fact that the two variables never differ by more than
2 is correct. Sadly, no idea for a proof (I don't suppose one
could prove the "differ by 2" statement by using Legendre's
theorem to count the multiplicity of prime factors of both terms?)

In any case, thanks for all your interest/time/computing power,
Stefan