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Generating Function 
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#1
May408, 10:10 PM

P: 13

1. The problem statement, all variables and given/known data
Find a linear homogeneous recurrence relation satisfied by an=2^n+n! 2. Relevant equations 3. The attempt at a solution The teacher gave us a hint using generating functions. The generating function for f(x) is f(x)=1+2x+4x^2+8x^3+....+1+x+2x^2+6x^3+24x^4+... The first part is a geometric series which equals 1/(12x), so f(x)=1/(12x)+1+x+2x^2+6x^3+24x^4+... Then he said to multiply both sides by x, take the derivative, and relate f'(x) to f(x). I found f'(x) after multiplying by x to be f(x)+xf'(x)=1/((12x)^2)+1+2x+6x^2+24x^3+... Now I don't know how to proceed. I need to relate the two, but am lost as to how to do that. Any suggestions? Thanks. 


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