SPOILER F: - - - - - - - - - - - - - - - - - - - - - - - - - - -

Solution: Bell Numbers

  B_n is the number of set partitions of the n-set.

  Example: There are 5 partitions of the set {1, 2, 3}.
       {{1,2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, {{1},{2},{3}}.

  The following difference relation is the easiest way to
  compute the first n terms.


  1     2     5    15    52   203   ...
     1     3    10    37   151
        2     7    27   114
           5    20    87
             15    67
                52
                  ...

Relations:

  B_n+1 = Sum_{i=0..n} B_i * BinomialCoefficient(n,i)  // Recursion
    
  B_n = S(n,0) + S(n,1) + ... + S(n,n)       // Sum of the Stirling 2nd Numbers

  exp(exp(x) - 1) = Sum_{n>=0} B_n * x^n / n!   // Generating Function


References:

[C1]  = L. Comtet, { Advanced Combinatorics}, Reidel, Dordrecht, Holland, 1974.
[RCI] = J. Riordan, { Combinatorial Identities}, Wiley, NY, 1968.