Improving calculations of the number of distinct alignments of two strings

In a recent work, Covington discusses the enumeration of two different sets of alignments of two strings of symbols using elementary combinatorial techniques. He defines two functions a(m, n) and A(m, n) to count the number of two-string alignments in his "small" and "middle" sets of alignments (respectively). He provides a recurrence for each of these functions which allows for the calculation of values of a(m,n) and A(m,n). In this note, we obtain generating functions for each of these functions. With the generating functions in hand, we provide improvements on Covington's recurrences, making the calculation of a(m, n) and A(m, n) much more efficient.