A Characterization of the bell numbers Original Research Article

Let Bn be the Bell numbers, and Ãn (n⩾0), B̃n (n⩾1) be the matrices defined by Ãn(i,j)=Bi+j (0⩽i,j⩽n), B̃n(i,j)=Bi+j+1 (0⩽i,j⩽n). It is shown that (Bn) is the unique sequence of real numbers such that det Ãn=det B̃n=n!! for all n, where n!!=∏k=0n(k!).