One to Integer Rising is Integer Factorial
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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 |
$\blacksquare$