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May 1st, 2010, 03:23 AM  #1  
Joined: May 2010 Posts: 3 Thanks: 0  Generating functions  (1xx^2x^3x^4x^5x^6)^1
Hi all! I've got this problem here: Quote:
I was able to solve this with recurrence relations: C0 = 1 (there is 1 way to roll a sum of 0), and CN = CN1 + CN2 + CN3 + CN4 + CN5 + CN6. 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.  
May 1st, 2010, 06:40 PM  #3 
Joined: May 2010 Posts: 3 Thanks: 0  Re: Generating functions  (1xx^2x^3x^4x^5x^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. 
May 3rd, 2010, 06:00 AM  #4 
Joined: Jun 2009 Posts: 150 Thanks: 0  Re: Generating functions  (1xx^2x^3x^4x^5x^6)^1
Do you know the power series expantion for in powers of Y ?

May 3rd, 2010, 08:47 AM  #5 
Joined: May 2010 Posts: 3 Thanks: 0  Re: Generating functions  (1xx^2x^3x^4x^5x^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. 

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