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

$\blacksquare$