Can you help me understand the beautiful formula for multifactorials?

[a1call]

Hi,
Can someone please decipher the following "beautiful" formula for me?

http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.htmlm(19)=1/19!+?+?+...
* What is the next/previous addend in the series sum?

Thanks in advance.

 

[jedishrfu]

I think there's a mistake in the MathWorld formula the summation of n=0 should be k=0

$m(n) = 1/( n! ) + 1/( n! * (n-1)! ) + 1/( n! * (n-1)! * (n-2)! ) + ...$

 

[a1call]

Thank you for your reply jedishrfu,
The article seems related to multifactorials with k being the number of exclamation marks, such as 2 marks for doublefactorial and more by similar extension.