 In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named for the Belgian mathematician Eugène Charles Catalan, 1814 to 1894. You could memorize that answer and say that you know what Catalan numbers are, but would that satisfy you? You might want to know how they are used and how do you find them, and then maybe if you can find a really big one, or what is the millionth Catalan number, if it even exists? So many questions, but first, a game involving strings of ones and zeros. The first rule is that strings must have equal numbers of ones and zeros. The second rule is that no initial segment has more zeros than ones. For n equals zero, there is one string, the empty string. No zeros, no ones, no initial segments, easy peasy. Next, n equals one. There are four strings, but only two will pass rule one. These two are either a zero followed by a one, or a one followed by a zero. But zero one fails rule two. So for n equals one, one zero is the only valid string. For n equals two, there are sixteen choices, but only six pass rule one with equal numbers of ones and zeros. We can also eliminate all strings that do not begin with one because of rule two. This leaves one one zero zero, one zero one zero, one zero zero one as the options. One zero zero one fails rule two further down the string. So for n equals two, there are only two valid strings. One one zero zero and one zero one zero. You get the idea.