After playing around in mathematica, I found that the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant $(-1)^r$ for the first few $r$'s.
How can I prove this this, or at the very least prove it in general has nonzero determinant? Are there are general methods to get a closed form for the inverse?
If $M$ is the $n \times n$ matrix with entries ${x-i \choose j-1}$, $i,j=1..n$, and $E$ the lower triangular matrix $n \times n$ with entries $(-1)^{i+j} {i-1 \choose j-1}$, then it seems that $E M$ is upper triangular with alternating $\pm 1$ on the diagonal. This boils down to some binomial identities.