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Find the generating function for the sequence $a_n$ which counts the number of bags containing $n$ pieces of fruit in which there is an odd number of apples, at most $1$ banana, and at least $1$ orange. Find $a_{40}$.

I came up with the equation $(x+x^3+x^5...)(1+x)(x+x^2+x^3+...)$ for the three fruits. Simplifying, it becomes $x^2(\frac{1}{1-x^2})(\frac{1-x^2}{1-x})(\frac{1}{1-x})$ and then to $\frac{x^2}{(1-x)^2}$.

Assuming my formula is correct, do I find $a_{40}$ by plugging $40$ into the $x$'s?

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No, $a_{40}$ is the coefficient of $x^{40}$ of the Taylor expansion of $\frac{x^2}{(1-x)^2}$. You can find this by starting $\frac{1}{1-x}=\sum_{n=0}^\infty x^n=1+x+x^2+\dots$ and taking the derivative to get $$\frac{1}{(1-x)^2}=\sum_{n=0}^\infty nx^{n-1}$$ So, $$\frac{x^2}{(1-x)^2}=\sum_{n=0}^\infty nx^{n+1}$$ What is the coefficient of $x^{40}$?

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  • $\begingroup$ if the exponent is $40$, would the coefficient be $39$? $\endgroup$ – LoudAnnoyance May 16 '19 at 15:33
  • $\begingroup$ Exactly, note that you can check this since you can actually count the number of combinations $(1,0,39),(1,1,38),(3,0,37),(3,1,36),\dots ,(37,0,3),(37,1,2),(39,0,1)$. $\endgroup$ – Julian Mejia May 16 '19 at 15:40

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