Enumeration of (k,2)(k,2)-noncrossing partitions

A set partition is said to be (k,d)(k,d)-noncrossing if it avoids the pattern 12⋯k12⋯d12⋯k12⋯d. We find an explicit formula for the ordinary generating function of the number of (k,d)(k,d)-noncrossing partitions of {1,2,…,n}{1,2,…,n} when d=1,2d=1,2.