Numerator of p-1th Harmonic Number is Divisible by Prime p/Proof 2

Proof
Note that for any integer $x$:

By comparing coefficients:
 * $\ds {p \brack p} \equiv 1 \pmod p$
 * $\ds {p \brack 1} \equiv -1 \pmod p$
 * $\ds {p \brack k} \equiv 0 \pmod p$ for $k \ne 1, p$

or in a more compact form:
 * $\ds {p \brack k} \equiv \delta_{k p} - \delta _{k 1} \pmod p$

where:
 * $\ds {p \brack k}$ denotes an unsigned Stirling number of the first kind
 * $\delta$ is the Kronecker delta.

From Harmonic Number as Unsigned Stirling Number of First Kind over Factorial:
 * $\ds H_{p - 1} = \frac {p \brack 2} {\paren {p - 1}!}$

From the above we have:
 * $\ds p \divides {p \brack 2}$

By Prime iff Coprime to all Smaller Positive Integers we also have:
 * $p \nmid \paren {p - 1}!$

Hence the numerator of $H_{p - 1}$, when expressed in canonical form, is divisible by $p$.