Divergent Power Series

Question

In "Generatingfunctionology", Wilf says that the divergent power series $A(x) = \sum_{k=0}^{\infty}k!x^k$ has many applications as a formal power series even though it is divergent. However, as far as I have seen he has not explicitly mentioned any of them, so what are those applications that he talks about? Are there any other divergent power series that end up useful as a formal power series?

Answer 1

There are three applications of $A(x)$ on page 373 0f Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick. (The book can be found online in pdf form.) Flajolet and Sedgewick use the notation $F(z)$ rather than $A(x)$.

(1) The number of permutations of size $n$ such that $\sigma_{i+1}-\sigma_i \ne 1$ has OGF $$F\left( \frac{z}{1+z} \right)$$

(2) The number of permutations of size $n$ such that $|\sigma_{i+1}-\sigma_i| \ne 1$ has OGF $$F \left( \frac{z(1-z)}{1+z} \right)$$

(3) The number of permutations of size $n$ such that $(\sigma_{i+1}-\sigma_i) \notin \{0,1,2\}$ has OGF $$\frac{1}{1-z^2} \left( -z + F \left( \frac{z(1-z)}{(1+z)(1+z-z^3)} \right) \right)$$