2
$\begingroup$

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:

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

$\endgroup$
3
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.