All Questions

Encouraged by David's proof for my earlier MO question, let's consider a similar problem. I can prove the below equality by computing each of the two sides, directly. That means, there is an ...

This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof. ...

The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define $$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$ Edit: Thanks to Johann ...

It is well known that for $n>0$ $$d(n)=\det\left(\binom{2i+2j+1}{i+j}\right)_{i,j=0}^{n-1}=1.$$ Computer experiments suggest that more generally $$d(n,k)=\det\left(\binom{2i+2j+2k+1}{i+j}\right)_{i,...