Balls on the Lawn

Abstract: In the "tennis ball" problem we are given successive pairs of balls numbered (1,2), (3,4),... At each stage we throw one ball out of the window. After n stages some set of n balls is on the lawn. We find a generating function and a closed formula for the sequence 3,23,131,664,3166,14545,65187,287060,1247690,..., the n-th term of which gives the sum over all possible arrangements of the total of the numbers on the balls on the lawn. The problem has connections with "bicolored Motzkin paths" and the ballot problem.