Recurrences for Alternating Sums of Powers of Binomial Coefficients

Abstract Define A r (n)=∑ n k=-n (-1) k ( 2n n+k ) r (r=2,3,...) . An elementary method is given for finding a recurrence for A r ( n ) with [ (r+2) 2 ] terms. Using asymptotics it is proved that this is the minimum number of terms in a recurrence for A r ( n ) if r is a prime or a power of 2.