TITLE: Proving that a combinatorial sequence has no compact formula
QUESTION [8 upvotes]: Suppose we have a sequence $a_n$ given by some combinatorial formula, e.g. involving a sum of n terms (like ${n  \choose k}^{10}3^{-k}$ etc.). Sometimes it is plausible that there is no compact formula for the $a_n$, where one has to adopt a reasonable definition of "compact" (i.e. using a constant, independent of $n$, number of primitive operations). Are there any methods of proving that a certain sequence $a_n$ has no compact formula, in much the same way differential Galois theory allows one to prove that certain integrals are nonelementary? Of course this would relative the choice of "computational primitives" (factorial, ${n \choose k}$ etc.)

REPLY [14 votes]: Yes, there are. See the book A=B by Petkovsek, Wilf, and Zeilberger. I will quote from it:

[Petk91] is the Ph.D. thesis of Marko
  Petkovsek, in 1991. In it he
  discovered the algorithm for deciding
  if a given recurrence with polynomial
  coeﬃcients has a "simple" solution,
  which, together with the algorithms
  above, enables the automated discovery
  of the simple evaluation of a given
  deﬁnite sum, if one exists, or a proof of nonexistence, if none exists
  (see Chapter 8). A deﬁnite
  hypergeometric sum is one of the form
  $f(n) = \sum_{k=-\infty}^\infty
> F(k,n)$, where $F$ is hypergeometric.