The Schröder number (S_n) is given by the formula (S_n = \frac{1}{n+1}\binom{2n}{n}).

Using the formula (S_n = \frac{1}{n+1}\binom{2n}{n}), we have (S_5 = \frac{1}{6}\binom{10}{5} = \frac{1}{6}\cdot 252 = 42).

The Schröder number (S_n) is related to the Catalan number (C_n) by the formula (S_n = C_n + 1).

The generating function for the Schröder numbers is (F(x) = \frac{1}{1-x-x^2}).

The asymptotic behavior of the Schröder numbers is given by (S_n \sim \frac{1}{\sqrt{n}}\left(\frac{4}{3}\right)^n).

The number of ways to triangulate a convex (n)-gon is given by the Schröder number (S_n).

The number of ways to parenthesize a product of (n) factors is given by the Catalan number (C_n).

The number of ways to arrange (n) distinct objects in a circle is given by the Schröder number (S_n).

The number of ways to color the faces of a cube with (k) colors, such that no two adjacent faces have the same color, is given by the Schröder number (S_k).

The number of ways to partition a set of (n) elements into (k) non-empty subsets is given by the Schröder number (S_k).

The number of ways to place (n) non-attacking rooks on an (n \times n) chessboard is given by the Schröder number (S_n).

The number of ways to construct a binary tree with (n) internal nodes is given by the Catalan number (C_n).

The number of ways to triangulate a convex (n)-gon with (k) interior points is given by the Schröder number (S_{n+k}).

The number of ways to partition a set of (n) elements into (k) non-empty subsets, such that each subset contains at least (2) elements, is given by (S_k - C_k).

The number of ways to arrange (n) distinct objects in a row, such that no two adjacent objects are the same, is given by the Catalan number (C_n).