Recurrences for alternating sums of powers of binomial coefficients (Q1801787)

Let \(A_ r(n)=\sum^ n_{k=-n}(-1)^ k{2n \choose n+k}^ r\) \((r=2,3,\dots)\). A recurrence for \(A_ r(n)\) with \(\left[{r+2 \over 2}\right]\) terms is obtained by an elementary method. 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. Let \(S_ r(n)=\sum^ n_{k=0}{n \choose k}^ r\). No lower bounds are known for the minimum number of terms in the recurrence for \(S_ r(n)\). Explicit recurrences for \(A_ r(n)\) for \(r=2,3,4,5,6\) and 7 are listed in the appendix.