The following generalization of a determinant formula proved by \textit{W. H. Mills}, \textit{D. P. Robbins} and \textit{H. Rumsey} jun. [Discrete Math. 67, 43-55 (1987; Zbl 0656.05006)] is proved. Let \(\Delta_ 0(u)=2\) and for \(j>0\) let \[ \Delta_{2j}(u)={(u+2j+2)_ j({1\over 2}u+2j+{3\over 2})_{j-1}\over(j)_ j\cdot({1\over 2}u+j+{3\over 2})_{j-1}} \] where \((A)_ j=A\cdot(A+1)\cdots(A+j-1)\). The main result is that, if \[ M_ n(x,y)=\text{det}\left({i+j+x\choose 2i-j}+{i+j+y\choose 2i- j}\right)_{0\leq i,j\leq n-1} \] and \[ N_ n(x,y)=\text{det}\left({2\over x+1-y}\left\{{i+j+x+1\choose 2i-j+1}-{i+j+y\choose 2i- j+1}\right\}\right)_{0\leq i,j\leq n-1} \] then \[ M_ n(x,y)=N_ n(x,y)=\prod^{n-1}_{k=0}\Delta_{2k}(x+y). \]