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December 29th, 2011, 07:39 AM   #1
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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.
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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.
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December 29th, 2011, 01:15 PM   #3
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Re: confusing

thanks
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