A Characterization of the bell numbers

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