Factorial Series Written As Recursive Definition

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I have a factorial series as shown below:

\begin{equation} (2n+1)!~\text{for all $n \geq 0$} \end{equation}

And I would like to know if the recursive definition that I wrote is accurate:

\begin{equation} Factorial(n)=\begin{cases} n, & \text{if $n<0$}.\\ (2n+1)\cdot Factorial((2n+1)-1), & \text{otherwise}. \end{cases} \end{equation}

Can anybody verify if I have done this correctly? Thanks a lot!

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1 Answer

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If you take $Factorial(n)$ for $n \geq 0$ according to the definition, this will be defined as:

$Factorial(n) = (2n+1)Factorial(2n) = (2n+1)(4n+1)Factorial(4n) = (2n+1)(4n+1)(8n+1)Factorial(8n) = \ldots$

I think what you may be looking for instead is:

$Factorial(n) = (2n+1)(2n)Factorial(n-1)$

Also, you may want to specify what $Factorial(n)$ is for some small values of $n$, since otherwise it's not clear what $Factorial(0)$ is for example.

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