# Harmonic Number as Unsigned Stirling Number of First Kind over Factorial

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## Theorem

- $H_n = \dfrac { {n + 1} \brack 2} {n!}$

where:

- $H_n$ denotes the $n$th harmonic number
- $n!$ denotes the $n$th factorial
- $\ds { {n + 1} \brack 2}$ denotes an unsigned Stirling number of the first kind.

## Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

- $H_n = \dfrac { {n + 1} \brack 2} {n!}$

$\map P 0$ is the case:

\(\ds H_0\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {1 \brack 2} {0!}\) | Unsigned Stirling Number of the First Kind of Number with Greater |

Thus $\map P 0$ is seen to hold.

### Basis for the Induction

$\map P 1$ is the case:

\(\ds H_1\) | \(=\) | \(\ds 1\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {2 \brack 2} {1!}\) | Unsigned Stirling Number of the First Kind of Number with Self |

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

- $H_k = \dfrac { {k + 1} \brack 2} {k!}$

from which it is to be shown that:

- $H_{k + 1} = \dfrac { {k + 2} \brack 2} {\paren {k + 1}!}$

### Induction Step

This is the induction step:

\(\ds \dfrac { {k + 2} \brack 2} {\paren {k + 1}!}\) | \(=\) | \(\ds \dfrac {\paren {k + 1} { {k + 1} \brack 2} + { {k + 1} \brack 1} } {\paren {k + 1}!}\) | Definition of Unsigned Stirling Numbers of the First Kind | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\paren {k + 1} { {k + 1} \brack 2} + k!} {\paren {k + 1}!}\) | Unsigned Stirling Number of the First Kind of n+1 with 1 | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\paren { {k + 1} \brack 2} } {k!} + \dfrac 1 {k + 1}\) | simplifying | |||||||||||

\(\ds \) | \(=\) | \(\ds H_k + \dfrac 1 {k + 1}\) | Induction Hypothesis | |||||||||||

\(\ds \) | \(=\) | \(\ds H_{k + 1}\) | Definition of Harmonic Number |

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

- $\forall n \in \Z_{\ge 0}: H_n = \dfrac { {n + 1} \brack 2} {n!}$

$\blacksquare$

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $6$