Factorial coefficient simplification

[physicsjock]

Hey,

I've been trying to find a way to simplify these two series coefficients,

$\begin{align} & {{a}_{4n}}={{a}_{o}}\frac{2!6!...(n-6)!(n-2)}{4!8!...(n-4)!n!} \\ & {{a}_{4n+1}}={{a}_{1}}\frac{3!7!...(n-6)!(n-2)}{5!9!...(n-4)!n!} \\ & E.g. \\ & {{a}_{12}}={{a}_{o}}\frac{2!6!10!}{4!8!12!}\,= \frac{{{a}_{o}}}{3\cdot 4\cdot 7\cdot 8\cdot 11\cdot 12}\,\,\,\,\,\,and \\ & {{a}_{13}}={{a}_{1}}\frac{3!7!11!}{5!9!13!}\,=\,\,\frac{{{a}_{1}}}{4\cdot 5\cdot 8\cdot 9\cdot 12\cdot 13} \\ \end{align}$

This is the best way I could find but I was wondering if there were any ways to avoid the use of ...'s

Is there any way to simplify these coefficients?

Thanks in advance

 

[mighty2000]

!

Hi there,

I can definitely help you with simplifying these series coefficients. It's great that you're trying to find a simpler way to represent them without using the ...'s. Let's take a closer look at the expressions you have provided:

\begin{align}
& {{a}_{4n}}={{a}_{o}}\frac{2!6!...(n-6)!(n-2)}{4!8!...(n-4)!n!} \\
& {{a}_{4n+1}}={{a}_{1}}\frac{3!7!...(n-6)!(n-2)}{5!9!...(n-4)!n!} \\
\end{align}

One way to simplify these coefficients is by using the factorial notation, denoted by the exclamation mark (!). For example, instead of writing 2!6!, we can write it as 2x1x6x5x4x3x2x1. This makes it easier to see the pattern and simplify the expressions.

Now, let's take a look at the first expression. We can rewrite it as:

\begin{align}
{{a}_{4n}}={{a}_{o}}\frac{(n-2)!}{4!(n-4)!n!}
\end{align}

Notice that the terms in the numerator and denominator are very similar. We can further simplify this by cancelling out the common terms, resulting in:

\begin{align}
{{a}_{4n}}={{a}_{o}}\frac{1}{4n(4n-1)(4n-2)(4n-3)}
\end{align}

Similarly, we can simplify the second expression as:

\begin{align}
{{a}_{4n+1}}={{a}_{1}}\frac{(n-2)!}{5!(n-4)!n!}={{a}_{1}}\frac{1}{5n(5n-1)(5n-2)(5n-3)}
\end{align}

This way, we have avoided using the ...'s and simplified the expressions to a great extent. I hope this helps you in your research. Keep up the good work!