GAP Manual: 46.3. Bell

Bell( n )

Bell returns the Bell number B(n). The Bell numbers are defined by B(0)=1 and the recurrence B(n+1) = sum_(k=0)^(n)((n choosek)B(k)).

B(n) is the number of ways to partition a set of n elements into pairwise disjoint nonempty subsets (see PartitionsSet). This implies of course that B(n) = sum_(k=0)^(n)(S_2(n,k)) (see Stirling2). B(n)/n! is the coefficient of x^n in the formal series e^(e^x-1), which is the generating function for B(n).

    gap> List( [0..6], n -> Bell( n ) );
    [ 1, 1, 2, 5, 15, 52, 203 ]
    gap> Bell( 14 );
    190899322