Catalan number

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 after the Belgian mathematician Eugène Charles Catalan (1814–1894). The nth Catalan number is given directly in terms of binomial coefficients by The first Catalan numbers for n = 0, 1, 2, 3, ... are An alternative expression for Cn is which is equivalent to the expression given above because ( 2 n n + 1 ) = n n + 1 ( 2 n n ) {displaystyle { binom {2n}{n+1}}={ frac {n}{n+1}}{ binom {2n}{n}}} . This shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations