Asymptotics of a family of binomial sums

Using a recent method of Pemantle and Wilson, we study the asymptotics of a family of combinatorial sums that involve products of two binomial coefficients and include both alternating and non-alternating sums. With the exception of finitely many cases the main terms are obtained explicitly, while the existence of a complete asymptotic expansion is established. A recent method by Flajolet and Sedgewick is used to establish the existence of a full asymptotic expansion for the remaining cases, and the main terms are again obtained explicitly. Among several specific examples we consider generalizations of the central Delannoy numbers and their alternating analogues.