Let $l$ be a positive integer. Does the matrix
$$
 M\_l \ := \ \left( \binom{l-(2p+1)}{j} \right)\_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?



Invertibility of a matrix whose entries are certain binomial coefficients
There is a nice result of Gessel and Viennot that computes your determinant in terms of NE lattice paths. The original paper is available [here](http://www.sciencedirect.com/science/article/pii/0001870885901215). Aigner and Ziegler also give a nice exposition of this result in their [Proofs From the Book](http://rads.stackoverflow.com/amzn/click/3642008550).


