Consider the sequence $ (a_{n})_{n \in \mathbb{N}} $ of positive integers whose first few entries are
$ 2 ~~ 6 ~~ 20 ~~ 70 ~~ 252 ~~ \ldots $
Now, consider the infinite matrix
\begin{equation} \left[ \begin{array}{cc} 1 & 1 & 1 & 1 & 1 & 1 & \cdots \\ 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\ 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 1 & 4 & 10 & 20 & 35 & 56 & \cdots \\ 1 & 5 & 15 & 35 & 70 & 126 & \cdots \\ 1 & 6 & 21 & 56 & 126 & 252 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \right]. \end{equation}
The $ (i,j) $-entry of this matrix indicates the number of ways of traveling from the $ (1,1) $-entry to the $ (i,j) $-entry of an $ (n \times n) $-matrix by only moving either right or down.
The sequence $ (a_{n})_{n \in \mathbb{N}} $ is formed from the diagonal elements of this matrix, starting from the $ (2,2) $-entry.
Question: How does one generate the $ n $-th entry of the sequence without referring to the matrix above? Is there a generating function for the sequence?