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May 13th, 2009, 05:33 AM  #1 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Factorial expressed as the sum of a series  new ?
I noted this result when I was a child If you take the series 1^n , 2^n, 3^n ........ Then repeatedly take the differences, after n operations you end up with n! I.e for n=2 then 1,4,9,16,25 3,5,7,9 2,2,2 n=3 1,8.27.64,125, 216 7,19,37,61,91 12,18,24,30 6,6,6 From this by considering the general series 1^n, 2^n, 3^n,........r^n...and taking differences I derived the general result obtaining the alternating series n! = sum (r = 0 to n) (1)^r * nCr * (n+1r)^n Where C is the combinatorial function nCr = n!/r!(nr)! or sum(r= 0 to n) (1)^r * (1/(r!(nr)!) * (n+1r)^n = 1 for any n i.e 5! = 6^5  5 * 5^5 + 10 * 4^5  10 * 3^5 + 5 * 2^5 1 = 7776  15625 + 10240  2430 + 160 1 = 120 Is this result known about elsewhere or am I just showing something trivial like n! = n! ? 
May 13th, 2009, 08:50 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Factorial expressed as the sum of a series  new ?
You've discovered an important result: . . . 
May 13th, 2009, 09:46 AM  #3 
Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0  Re: Factorial expressed as the sum of a series  new ?
That last one is nice. Here it is in LaTeX (more readable) 
May 14th, 2009, 01:36 AM  #4  
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: Factorial expressed as the sum of a series  new ? Quote:
Given n+x boxes and n balls, how much different ways there're to put the n balls into the n+x boxes and none of the first n boxes is empty? Using InclusionExclusion Principle, the result is the leftside of the equation. And it is obvious the result is n! too.  
May 14th, 2009, 02:29 AM  #5 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Re: Factorial expressed as the sum of a series  new ?
How can I find out if this series result has been obtained before, as seems fairly likely ? And more importantly why would we expect an alternating set of powers of n+1 > 1 raised to the n to produce n! 
October 31st, 2009, 02:29 AM  #6  
Newbie Joined: Oct 2009 Posts: 1 Thanks: 0  Re: Factorial expressed as the sum of a series  new ? Quote:
This is, for me, less clear...  

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