Divisibility and Binomial Coefficients

[Stefan Steinerberger]

Dear seqfans,

JM Bergot has asked me whether there are a < m,n
such that C(m, a) + C(n, a) divides C(m+n,a), where
C(m,n) is the binomial coefficient. As the nature of the
problem suggests, there is a vast number of solutions
(for example C(4,3) + C(5,3) = 14, C(9,3) = 6*14).

I find the "dual" question more interesting. Are there
distinct a,b <= n such that C(n,a) + C(n,b) divides C(n,a+b)?
I only found the following seven solutions

C(19,3)+C(19,5) | C(19,8)
C(34,6)+C(34,7) | C(34,13)
C(41,5)+C(41,7) | C(41,12)
C(89,7)+C(89,8) | C(89,15)
C(104,3)+C(104,4) | C(104,7)
C(359,5)+C(359,6) | C(359,11)
C(398,20)+C(398,21) | C(398,41)

Is there any reason to assume that the number of solutions is
finite/infinite?

Best wishes,
Stefan