Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing   Almkvist-Zeilberger Algorithm]
  Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of them is both famous and not-yet-proved-irrational. 
Shalosh B. Ekhad and Doron Zeilberger
math.NT