7:53 AM
But if there is a question with a totally non-descriptive title, nobody edits the title to include the formula in question. (Which would also remove the question from HNQ.)
3

Define the sequence given by the finite sum $$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$ Questions. (1) Is $0<a_n<1$? (2) Does the limit $\lim_{n\rightarrow\infty}a_n$ exist? If yes, what is its value?