The "Beautiful" Formula

[a1call]

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

http://mathworld.wolfram.com/Recipro...lConstant.html


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

Thanks in advance.

[a1call]

Originally Posted by a1call

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

http://mathworld.wolfram.com/Recipro...lConstant.html


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

Thanks in advance.

ETA
I just realized that I put an example with too many Exclamation marks for n=3.

[Nick]

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

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

[a1call]

Originally Posted by Nick

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

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

Thank you for the reply Nick,
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?

[VBCurtis]

Originally Posted by a1call

Can't wrap my head around what happens when the Exclamation marks exceed n.

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

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.

[a1call]

Originally Posted by VBCurtis

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

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.

Thank you VBCurtis,
So the only question marks remaining is:
* What is the definition of a multifactorial of n with 0 exclamation marks
* 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 VBCurtis

[science_man_88]

Originally Posted by VBCurtis

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

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.

I think the problem is that the first part of the ink he gave says where what's being summed contains a value k not given and n is an input then varies.

[a1call]

If this code is complete and all inclusive:

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

fac(19,0)

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

[VBCurtis]

Originally Posted by a1call

* What is the definition of a multifactorial of n with 0 exclamation marks

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.

[a1call]

Originally Posted by VBCurtis

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

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

n=19
--------

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

[science_man_88]

Originally Posted by a1call

Well a n would be out of phase

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.