One to Integer Rising is Integer Factorial

Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

$1^{\overline n} = n!$

where:

$1^{\overline n}$ denotes the rising factorial
$n!$ denotes the factorial.


Proof

\(\ds 1^{\overline n}\) \(=\) \(\ds \dfrac {\paren {1 + n - 1}!} {\paren {1 - 1}!}\) Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \dfrac {n!} {0!}\)
\(\ds \) \(=\) \(\ds n!\) Factorial of Zero

$\blacksquare$

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