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December 24th, 2011, 03:08 AM   #1
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nth term

If I have generating function, nth term of its series is given by formula: c_n=(f^(n) (0) )/n!
Problem is f^(n), nth derivative of generating function, specially if generating function is quotient of two functions.
Is there a formula for nth term of series that will not be difficult to calculate with it?
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December 24th, 2011, 12:37 PM   #2
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Re: nth term

It's very easy to take the derivative of a gf -- just remove the constant term, multiply all remaining terms by their exponents, and reduce exponents by 1. The generalization to second, ... derivatives is easy; you can write out the multiple using the factorial function if you like.
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December 25th, 2011, 11:40 AM   #3
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Re: nth term

i.e. generating function is f(x)=(sum_(k=1,to m-1) x^k)/(1-x^m), where m is period;
0,1,1,0,1,1... m=3
0,1,1,1,0,1,1,1,0... m=4...

What is nth of the series given by the generating function?
Formula must be general, so I can just put m.
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