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Triangle of the Stirling Numbers of the First Kind

The Stirling numbers of the first kind $\left[\begin{array}{c}n\\r\end{array}\right],$ where $n$ and $r$ are natural numbers, are named after James Stirling (1692 – 1770). According to the corresponding recursive formula, they form a triangular scheme, in analogy to the Pascal’s triangle for binomial coefficients. For the first $10$ values of $n$ this scheme is

\[\begin{array}{r|rrrrrrrrrr}
n&\left[\begin{array}{c}n\\0\end{array}\right]&\left[\begin{array}{c}n\\1\end{array}\right]&\left[\begin{array}{c}n\\2\end{array}\right]&\left[\begin{array}{c}n\\3\end{array}\right]&\left[\begin{array}{c}n\\4\end{array}\right]&\left[\begin{array}{c}n\\5\end{array}\right]&\left[\begin{array}{c}n\\6\end{array}\right]&\left[\begin{array}{c}n\\7\end{array}\right]&\left[\begin{array}{c}n\\8\end{array}\right]&\left[\begin{array}{c}n\\9\end{array}\right]&\left[\begin{array}{c}n\\10\end{array}\right]\\
\hline
0&1\\
1&&1\\
2&&1&1\\
3&&2&3&1\\
4&&6&11&6&1\\
5&&24&50&35&10&1\\
6&&120&274&225&85&15&1\\
7&&720&1764&1624&735&175&21&1\\
8&&5040&13068&13132&6769&1960&322&28&1\\
9&&40320&109584&118124&67284&22449&4536&546&36&1\\
10&&362880&1026576&1172700&723680&269325&63273&9450&870&45&1\\
\end{array}\]

Note that empty entries in this table are actually \(0\)’s.

| | | | created: 2020-04-04 12:29:03 | modified: 2020-04-10 18:35:30 | by: bookofproofs | references: [8404]

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Bibliography (further reading)

[8404] Miller, Kenneth S.: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 1960