In my spare hours I sometimes enjoy playing a game called "WTF, The On-Line Encyclopedia of Integer Sequences?". It goes like this.
Step one: think of a combinatorial problem. Such as, count the graph-like structures like
| / \ / \ | | / \ | \ / | | | | / \ | \ / | | \ / \ / |
where, viewing this as a Feynman-diagram-like history of a little particle, the particle splits up and joins back up a few times.
Step two: figure out a recurrence well enough to crank out a few terms manually. Such as, the number of tree structures above that begin and end with one particle, and which have 0, 2, 4, 6, 8 interactions are 1,1,5,61,1385. (This is ignoring "concurrency" because I don't know how to efficiently count them otherwise --- that is, the diagram above and its mirror image are considered different, and indeed there are a couple other interleavings of the duplication and joining of the two "inner" particles)
Step three: Look that shit up on TOLEoIS.
Step four: WTF, The On-Line Encyclopedia of Integer Sequences? It's the coefficients of the power series for the secant function? Why the heck is that true?