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 December 29th, 2011, 07:39 AM #1 Newbie   Joined: Nov 2011 Posts: 20 Thanks: 0 confusing I have series: 0,1,0,1,0,1,... For example I have to get fourth number in the series, but with generating function. The generating function is: 0+1x + 0x^2 + 1x^3 + 0x^4+... =(1+x^2 + x^3 + x^4...)-(1 + x^2 + x^4 +...)=1/(1-x) - 1/(1-x^2) The formula for getting nth number in the series is given by formula: f(n)= (f^(n) (0))/n! , where f(x) must be n times derivatived, and then x=0 I did that: (1/(1-x) - 1/(1-x^2))''''=-(24 x (5+10 x^2+x^4))/(-1+x^2)^5 ___(wolfram alpha) I put 0 instead x, and I get f(n)=0/n! = 0 but I should get 1. What is the problem, is the formula wrong or something else? *I know there exist a lot of simpler formulas, but if I have series 0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1... generating function is powerful tool, and I don't have to use floor, ceiling or modulo functions.
December 29th, 2011, 10:40 AM   #2
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Re: confusing

Quote:
 Originally Posted by Emilijo I have series: 0,1,0,1,0,1,... For example I have to get fourth number in the series, but with generating function. The generating function is: 0+1x + 0x^2 + 1x^3 + 0x^4+... =(1+x^2 + x^3 + x^4...)-(1 + x^2 + x^4 +...)=1/(1-x) - 1/(1-x^2) The formula for getting nth number in the series is given by formula: f(n)= (f^(n) (0))/n! , where f(x) must be n times derivatived, and then x=0 I did that: (1/(1-x) - 1/(1-x^2))''''=-(24 x (5+10 x^2+x^4))/(-1+x^2)^5 ___(wolfram alpha) I put 0 instead x, and I get f(n)=0/n! = 0 but I should get 1. What is the problem, is the formula wrong or something else?
Why are you going through all of that? If you want term 3 (the fourth term starting from 0), just look at the coefficient of x^3. If you take the derivative 3 times, substitute x = 0, and divide the result by 3! you'll get the same result, but the whole point of the generating function is that the sequence is the same as the coefficients.

 December 29th, 2011, 01:15 PM #3 Newbie   Joined: Nov 2011 Posts: 20 Thanks: 0 Re: confusing thanks

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