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May 1st, 2010, 03:23 AM   #1
 
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Generating functions - (1-x-x^2-x^3-x^4-x^5-x^6)^-1

Hi all!

I've got this problem here:
Quote:
Show that (1 - x - x^2 - x^3 - x^4 - x^5 - x^6)^-1 is the generating function for the number of ways a sum of r can occur if a die is rolled any number of times.
It's a homework problem, but it's past due, so I am just trying to figure it out because I think I may have missed something important that I need to solve this.

I was able to solve this with recurrence relations:
C0 = 1 (there is 1 way to roll a sum of 0), and CN = CN-1 + CN-2 + CN-3 + CN-4 + CN-5 + CN-6.
I used that to get h(x) - x - 2x^2 - 4x^3 - 8x^4 - 16x^5 - 32x^6 = x (h(x) - x - 2x^2 - 4x^3 - 8x^4 - 16x^5) + x^2 (h(x) - x - 2x^2 - 4x^3 - 8x^4) + x^3 (h(x) - x - 2x^2 - 4x^3) + x^4 (h(x) - x - 2x^2) + x^5 (h(x) - x) + x^6 (h(x)), where g(x) = h(x) + 1, because h(x) does not include the 1 * x^0 term. This works out fine.

But there should be some way to verify this without recurrence relations. I'm not sure how to convert the generating function to a sum, or how to reconstruct (build) the generating function, but there should be some way to do either of those.

Hopefully someone here can give some advice.
Thanks.
KristopherWindsor is offline  
 
May 1st, 2010, 01:35 PM   #2
 
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Re: Generating functions - (1-x-x^2-x^3-x^4-x^5-x^6)^-1

Taylor expansion

(Sloane's A001592)

Xitami is offline  
May 1st, 2010, 06:40 PM   #3
 
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Re: Generating functions - (1-x-x^2-x^3-x^4-x^5-x^6)^-1

Thanks for the reply!

Now I understand that this is the generating function for the problem:

And I understand that it expands like this:


But I'm not quite sure how to convert between that and .
Could you please explain that?
Thanks.
KristopherWindsor is offline  
May 3rd, 2010, 06:00 AM   #4
 
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Re: Generating functions - (1-x-x^2-x^3-x^4-x^5-x^6)^-1

Do you know the power series expantion for in powers of Y ?
g_edgar is offline  
May 3rd, 2010, 08:47 AM   #5
 
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Re: Generating functions - (1-x-x^2-x^3-x^4-x^5-x^6)^-1

Well, I understand this:


But I'm not sure how it relates to , , etc.
Again, I'm not exactly sure what I'm missing here. :S

I'm not sure how to show this:


It'd be nice if you could explain that.
KristopherWindsor is offline  
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