# General Expression for an Element of a Triangular Array

The Pascal's Triangle is formed by the following rules:

$i,j \in N^*$

$T(1,1) = 1$

$\forall j>i\;\;\; ; \;\;\;T(i,j) = 0$

$\forall i>1\;\;\; ; \;\;\;T(i,j) = T(i-1,j-1) + T(i-1,j)$

where $i$ is the row index and $j$ is the column index

There is a general formula for its non-null terms:

$T(i,j) = \left(\begin{array}{c}i\\j\end{array}\right) = \frac{i!}{j!(i-j)!}$

I would like to know if there is a general expression for the terms of the following triangular array:

$i,j \in N^*$

$T(1,1) = 1$

$\forall j>i\;\;\; ; \;\;\;T(i,j) = 0$

$\forall i>1\;\;\; ; \;\;\;T(i,j) = T(i-1,j-1) + j\;T(i-1,j)$

Here the five first rows:

$\begin{bmatrix}1&0&0&0&0\\ 1&1&0&0&0\\ 1&3&1&0&0\\ 1&7&6&1&0\\ 1&15&25&10&1 \end{bmatrix}$

These number are ${i \brace j}$, the Stirling numbers of the Second Kind. They are archived as A008277 in OEIS.org.