I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just managed to reduce a problem to the question in the title, namely: given an integer $k$, what is the smallest $n$ such that 


$${n \choose \lfloor n/2\rfloor } > k$$


This term ${n \choose \lfloor n/2\rfloor }$ comes up many places. Obviously, it's the max of the $n$ choose $r$ function. It's also the conclusion of Sperner's Theorem on antichains, and the paper *A Sperner Theorem on Unrelated Chains of Subsets* by Griggs-Stahl-Trotter relates it to the 2-dimension of a disjoint union of posets. But I can't seem to find a formula anywhere for $n$ as a function of $k$. I really want a precise formula, but currently don't even have a proof of the asymptotics so that could be a place to start. 


Sorry if this is too elementary. This question has proven difficult to Google, because of its symbolic nature.



Given k, what is the minimum n such that n choose n/2 is greater than k?
This [note by Kessler and Schif](http://u.math.biu.ac.il/~schiff/Papers/prepap3.pdf)f gives pretty extensive asymptotics for the central binomial coefficients. I am quite sure that any sort of exact formula is hopeless.


