Abstract We study the asymptotic behavior of the terms in sequences satisfying recur- rences of the form an = anư 1 + P nư d k=d f (n, k)akanư k where, very roughly speaking, f (n, k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlev´e I equations are discussed in detail.