# Invertibility of a matrix whose entries are certain binomial coefficients

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?

• Have you searched for "Vandermonde determinant"? – Jason Starr Feb 21 '15 at 16:32
• Yes, but Vandermonde determinant didn't solve the problem. I found a paper that solves the problem but to a much more general case. I would like a more simpler solution, I believe it exists. – Liliam Feb 21 '15 at 16:40
• @JasonStarr How do you reduce this to Vandermonde? Just curious :) – Alex Degtyarev Feb 21 '15 at 16:50
• @AlexDegtyarev. For elements $a_0,\dots,a_{n-1}$ in a commutative ring $R$, the Vandermonde determinant $\text{det}(a_i^j)_{0\leq i,j\leq n-1}$ equals $0!\cdot 1!\cdots (n-1)!$ times the determinant $\text{det}(\binom{a_i}{j})_{0\leq i,j\leq n-1}$. Thus, the determinant above is, up to a product of factorials, a Vandermonde determinant. – Jason Starr Feb 21 '15 at 17:53
• @JasonStarr: thanks, this is nice. For proof, I guess, this is the interpolation by Newton "monomials" rather than conventional ones, right? – Alex Degtyarev Feb 21 '15 at 18:52

• @Liliam, I think if you take $k=\left\lfloor\frac{\ell-1}{2}\right\rfloor$, $a_i=2i$ and $b_j=j$ in their Theorem 1, then you get something pretty close to your determinant. – Chris McDaniel Feb 21 '15 at 18:12