I was curious if anyone has a reference for a formula giving the values of n and k so that $\binom{n}{k}<\binom{n+j}{k-1}$ for a fixed $j$.
Clearly this will be true if $k>\frac{n}{2}$ because then one will have that $\binom{n}{k}\le\binom{n}{k-1}<\binom{n+j}{k-1}$. One can improve on this result, and in the case where $j=1$ I have found precise conditions on $k$ in terms of $n$, but my approach is rather blunt and it seems like the general case will be quite tedious using my methods, even though this seems like a question that likely has an elegant combinatorial solution. I was wondering if anyone knows where a solution appears.