A certain pattern is $3\times 7\times 11\times \ldots \times(4n-1)$. How could this pattern be written as a factorial? There doesn't seem to be a way to factor out some number but is there a possible way to add one to each number?
I understand that patterns such as $2\times 4\times 6\times \ldots \times (2n)$ can be written like $2^n \times n!$. Is this also possible for this pattern? I can't seem to find a way to write it.
In terms of pure factorials, I do not think that this is possible.
However, if you remember that $\Gamma(x+1)=x!$, you can write $$\prod_{i=1}^{n} (4i-1)=4^n \,\frac{\Gamma \left(n+\frac{3}{4}\right)}{\Gamma \left(\frac{3}{4}\right)}=4^n \,\frac{\left(n-\frac{1}{4}\right)!}{\left(-\frac{1}{4}\right)!}$$