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Proposition: Recursive Formula for the Stirling Numbers of the First Kind

The Stirling numbers of the first kind obey the following recursive formula $$\left[\begin{array}{c}n+1\\r\end{array}\right]=\left[\begin{array}{c}n\\r-1\end{array}\right]+n\cdot \left[\begin{array}{c}n\\r\end{array}\right]$$ with the initial conditions $$\begin{align}\left[\begin{array}{c}n\\n\end{array}\right]&:=1,\quad n\ge 1\nonumber\\\left[\begin{array}{c}n\\r\end{array}\right]&:=0,\quad r=0 < n\text{ or }n < r.\nonumber\end{align}$$

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References

Bibliography

1. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960