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

## 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:

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

$\blacksquare$