# Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1

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## Definition

**Unsigned Stirling numbers of the first kind** are defined recursively by:

- $\displaystyle {n \brack k} := \begin{cases} \delta_{n k} & : k = 0 \text { or } n = 0 \\ & \\ \displaystyle {n - 1 \brack k - 1} + \paren {n - 1} {n - 1 \brack k} & : \text{otherwise} \\ \end{cases}$

where:

- $\delta_{n k}$ is the Kronecker delta
- $n$ and $k$ are non-negative integers.

## Notation

The notation $\displaystyle {n \brack k}$ for the unsigned Stirling numbers of the first kind is that proposed by Jovan Karamata and publicised by Donald E. Knuth.

The notation $\map s {n, k}$ for the signed Stirling numbers of the first kind is similar to variants of that sometimes given for the unsigned.

Usage is inconsistent in the literature.

## Also see

## Source of Name

This entry was named for James Stirling.

## Technical Note

The $\LaTeX$ code for \(\displaystyle {n \brack k}\) is `\displaystyle {n \brack k}`

.

The braces around the `n \brack k`

are **important**.

The `\displaystyle`

is needed to create the symbol in its proper house display style.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(46)$