Asymptotics of Some Convolutional Recurrences

Edward A. Bender, Adri B. Olde Daalhuis, Zhicheng Gao, L. Bruce Richmond, Nicholas Wormald


We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form $a_n = a_{n-1} + \sum_{k=d}^{n-d} f(n,k) a_k a_{n-k}$ 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 Painlevé I equations are discussed in detail.

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