What's the meaning of $\choose$ in this formula:
$$C_n=\frac1{n+1}\binom{2n}{n}=\frac{(2n)!}{n!(n+1)!}\qquad\mathrm{for}\;n\geq 0$$
Is it a division?
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The notation $n\choose r$ means "the number of ways of choosing $r$ things out of a total of $n$ given things, where order is not important." Its value is $n!/r!(n-r)!$.
For example, given five coins, say a penny, a nickel, a dime, a quarter, and a dollar, there are 10 ways to choose two coins with order unimportant, because ${5\choose 2} = 5!/2!3! = 120/(2)(6) = 10$.
Larry Denenberg
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3It might also be useful to note that it is referred to commonly as the Binomial Coefficient – Justin L. Oct 27 '10 at 16:15
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3And we might add that the way to pronounce $\binom{n}{r}$ is "$n$ choose $r$". – Arturo Magidin Oct 27 '10 at 18:29