Enumerating split-pair arrangements

An arrangement of the multi-set { 1 , 1 , 2 , 2 , … , n , n } is said to be “split-pair” if for all i < n , between the two occurrences of i there is exactly one i + 1 . We enumerate the number of split-pair arrangements and in particular show that the number of such arrangements is ( − 1 ) n + 1 2 n ( 2 2 n − 1 ) B 2 n where B i is the ith Bernoulli number.