The "Beautiful" Formula
 

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

[url]http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html[/url]


m(3)=1/3!!!!!+?+?+.....
* What's the next/previous addend in the series sum?

Thanks in advance.

[QUOTE=a1call;433784]Hi,
Can someone please decipher the following "beautiful" formula for me?

[url]http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html[/url]


m(3)=1/3!!!!!+?+?+.....
* What's the next/previous addend in the series sum?

Thanks in advance.[/QUOTE]
ETA
I just realized that I put an example with too many Exclamation marks for n=3.

I assume that they mean this (in your example):

\[m(3)=\frac{1}{3}+\frac{1}{3!}+\frac{1}{3!!}+\frac{1}{3!!!}+\ldots\]

[QUOTE=Nick;433799]I assume that they mean this (in your example):

\[m(3)=\frac{1}{3}+\frac{1}{3!}+\frac{1}{3!!}+\frac{1}{3!!!}+\ldots\][/QUOTE]

Thank you for the reply [B]Nick[/B],
So it should be k=0 rather than n=0 then.
Can't wrap my head around what happens when the Exclamation marks exceed n.
ETA
And k=0 is undefined, or is a multifactorial with 0 exclamation marks just n?

[QUOTE=a1call;433848]
Can't wrap my head around what happens when the Exclamation marks exceed n.
[/QUOTE]

Perhaps you should read the definition of multifactorial:
[url]http://mathworld.wolfram.com/Multifactorial.html[/url]

The examples given indicate what the result is for, say, 3!!!!. So does the definition, but for some folks examples are the way to enlightenment.

[QUOTE=VBCurtis;433861]Perhaps you should read the definition of multifactorial:
[url]http://mathworld.wolfram.com/Multifactorial.html[/url]

The examples given indicate what the result is for, say, 3!!!!. So does the definition, but for some folks examples are the way to enlightenment.[/QUOTE]
Thank you [B][B]VBCurtis[/B][/B],
So the only question [B]marks [/B]remaining is:
* What is the definition of a multifactorial of n with 0 exclamation [B]marks[/B]
* Does the "beautiful" formula has a typo showing n=0 (to my poor vision) instead of k=0 or while we are at it k=1

I appreciate your clarification [B][B]VBCurtis[/B][/B]:smile:

[QUOTE=VBCurtis;433861]Perhaps you should read the definition of multifactorial:
[url]http://mathworld.wolfram.com/Multifactorial.html[/url]

The examples given indicate what the result is for, say, 3!!!!. So does the definition, but for some folks examples are the way to enlightenment.[/QUOTE]

I think the problem is that the first part of the ink he gave says [TEX]m(n) = \sum_{n=0}^\infty ... [/TEX] where what's being summed contains a value k not given and n is an input then varies.

If this code is complete and all inclusive:

[CODE]fac(n,d)=prod(k=0,(n-1)\d,n-k*d);\\Multifactorial\\Credits: http://rosettacode.org/wiki/Multifactorial#PARI.2FGP

fac(19,0)
[/CODE]


[CODE] *** at top-level: fac(19,0)
*** ^---------
*** in function fac: prod(k=0,(n-1)\d,n-k*d)
*** ^---------
*** _\_: impossible inverse in sdivsi_rem: 0.
[/CODE]

[QUOTE=a1call;433868]
* What is the definition of a multifactorial of n with 0 exclamation [B]marks[/B]
[/QUOTE]

Are you really asking what "n" means? An "n" without a factorial symbol would be... n.

Also: what is 0! defined as? If you don't know, look it up.

[QUOTE=VBCurtis;433876]Are you really asking what "n" means? An "n" without a factorial symbol would be... n.

[/QUOTE]
Well a n would be out of phase in the sequence/series:

n=19
--------

n=19
n!=121645100408832000
n!!=654729075
n!!!=1106560
n!!!!=65835

[QUOTE=a1call;433879]Well a n would be out of phase[/QUOTE]

ah but if you define the value with 0 factorials behind it the same way as say n!:

n! = n*(n-1)*(n-2)... *2*1
n!! = n*(n-2)*(n-4)....

then continuing downward:
n = n*(n-0)*(n-2*0) you run into the problem that no matter how many times you subtract 0 you won't hit the negative numbers so is it n^oo ? that's what it would need to be to fit into the rest of the values.