1, 2, 5, 14, 42...

they describe the number of ways to legally place n parentheses, number of ways to triangulate a polygon with n+2 sides and plenty of other combinatorics problems...

C_n = (2n)!/n!(n+1)!

can also be computed by a

Pascal-like triangle:

1
1 1
1 2 2
1 3 5 5
1 4 9 14 14

where the catalan numbers are on the diagonal and each number is the sum of the number above it and to its left.
named after Eugene Charles Catalan (1814-1894).