Wondering if the numbers satisfying the following relationship have a name or known closed-form solution. They show up in enumerating possible configurations of swaps during the execution of a bubble sort.
\begin{equation} F_{k,n} = \begin{cases} 0 & \text{ k = 0} \\ k & \text{ n = 1} \\ F_{k-1,n} + F_{k+1,n-1} & \text{otherwise} \end{cases} \end{equation}
Note that $F_{1,n}$ is the nth Catalan number. The matrix of the first few numbers is:
\begin{array} 01&2&5&14\\ 2&5&14&42\\ 3&9&28&90\\ 4&14&48&165\ \end{array}
These are sometimes known as the ballot numbers. Imagine an election where candidate $A$ received $n+k$ votes, and candidate $B$ received $n$ votes. Then $F_{k,n}$ is the number of ways to order the ballots such that $A$ is strictly ahead of $B$ throughout the counting process when the ballots are counted in that order.
They are given by the explicit formula $$ F_{k,n}=\binom{2n+k}{n}\frac{k}{k+2n}. $$