Coins in a fountain

Summary

Coins in a fountain is a problem in combinatorial mathematics involving a generating function. In this problem, a fountain is an arrangement of non-overlapping coins, modeled as unit circles, into horizontal rows in the plane. Consecutive coins in the bottom row are tangent to each other, and each coin in a higher row is tangent to two coins in the row immediately below it. Above the bottom row, consecutive coins are not required to touch. The problem asks for the number of different fountains of $n$ coins with $k$ coins in the bottom row.