omg generating functions are just the most beautiful thing ever.
Imagine you have a model of a graph structure over time. Every time step a vertex is added, and with probability δ, also a random edge is added to the graph, with its endpoints uniform randomly chosen among the existing vertices. Imagine also that you're crazy and, even after you've satisfied the δ probability that you're going to try to add an edge, there's a further probability σ that you add a broken edge that just has two half-edges dangling from the two vertices it would have connected. (we actually have a motivation for this, though it sounds crazy)
What you can do is write down recurrences that describe just how many connected components you should expect to find, asymptotically, that have n vertices and m half-edges in them. These recurrences take up about half a page. Nested sums, the works.
Alternatively, you multiply all these guys by nxnym, sum them up, and like magic, all of the nested sums turn out to be what you would have gotten by doing algebraic and analytic manipulation of formal power series. In other words, that whole half-page of information is expressed perfectly well by a single line:
g = x + 2δgxx(-1 + σy + (1-σ)g)
That -1 there describes the fact that the number of n-sized m-half-edged (call them "(n,m)") components tends to decrease by the process of becoming other kinds of components; this happens when one end of an edge lands in one. The σy describes the process of a new (n,m+1) component being created by a half-edge being added to an (n,m) one. The (1-σ)g describes components joining together, and hides four nested summations ordinarily required to say everthing about the sizes and half-edges adding up right. The x on the left is the fact that we're adding a new singleton vertex (a (1,0) component) every time step. That's it! Math is so fucking beautiful sometimes.