I want to understand how I would go about solving counting problems like this
Use exponential generating functions to determine the number $a_n$ of ordered choices of $n$ balls such that there are 2 or 4 red balls, an even number of green balls, and an arbitrary number of blue balls
I understand how I would start solving this type of problem, getting a multiplication that starts like:
$(\dfrac{x^2}{2!} + \dfrac{x^4}{4!})(\dfrac{1}{2}(e^x + e^{-x}))(e^x)$
corresponding to the red, green, blue balls respectively.
I am aware of an existing question asking precisely this. However the answer there does not make sense to me question is here. Precisely speaking, I cannot understand how the multiplication of the generating functions is expanded in the first expansion in the answer there. Also please note that I cannot comment in the answer to the original question there, since I am a beginner and my reputation doesn't allow me to do that 🙂
Best Answer
If I understand your question correctly, you want to see the intermediate steps to show that if $$f(x)={\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)}{\frac12\left(e^x+e^{-x}\right)}{e^x}$$ then $$f(x)=\frac12\cdot\frac{x^2}{2!} + \frac12\cdot\frac{x^4}{4!} + \sum_{n=2}^\infty \frac1{16}2^nn(n-1)\frac{x^n}{n!} + \\\sum_{n=4}^\infty\frac1{768}2^nn(n-1)(n-2)(n-3)\frac{x^n}{n!}. \tag{1}$$ OK, here goes: $$\begin{align} f(x)&={\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)}{\frac12\left(e^x+e^{-x}\right)}{e^x} \\ &={\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)} \frac{1}{2}(e^{2x}+1) \\ &=\frac{1}{2}{\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)} + \frac{1}{2}{\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)}e^{2x} \\ &=\frac{1}{2}{\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)} + \frac{1}{2}{\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n \tag{2} \\ &=\frac{1}{2}{\left(\frac{x^2}{2!}+\frac{x^4}{4!}\right)} + \frac{1}{2} \cdot \frac{x^2}{2!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n + \frac{1}{2} \cdot \frac{x^4}{4!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n \\ &=\frac{1}{2} \cdot \frac{x^2}{2!}+\frac{1}{2} \cdot \frac{x^4}{4!} + g(x) + h(x) \tag{3} \end{align}$$ where for the sake of keeping our equations short we define $$g(x) = \frac{1}{2} \cdot \frac{x^2}{2!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n$$ and $$h(x) = \frac{1}{2} \cdot \frac{x^4}{4!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n$$ Then $$\begin{align} g(x) &= \frac{1}{2} \cdot \frac{x^2}{2!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n \\ &= \frac{1}{4} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^{n+2}\\ &= \frac{1}{4} \sum_{i=2}^{\infty} \frac{1}{(i-2)!} 2^{i-2} x^i \tag{4}\\ &= \frac{1}{4} \sum_{i=2}^{\infty} \frac{1}{(i-2)!} \cdot \frac{i(i-1)}{i(i-1)} \frac{2^i}{4} x^i\\ &= \frac{1}{16} \sum_{i=2}^{\infty} \frac{i(i-1)}{i!} 2^i x^i \tag{5} \end{align}$$ and $$\begin{align} h(x) &= \frac{1}{2} \cdot \frac{x^4}{4!} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^n\\ &= \frac{1}{48} \sum_{n=0}^{\infty} \frac{1}{n!} 2^n x^{n+4}\\ &= \frac{1}{48} \sum_{j=4}^{\infty} \frac{1}{(j-4)!} 2^{j-4} x^j \tag{6}\\ &= \frac{1}{48} \sum_{j=4}^{\infty} \frac{1}{(j-4)!} \cdot \frac{j(j-1)(j-2)(j-3)}{j(j-1)(j-2)(j-3)} \frac{2^j}{16} x^j\\ &= \frac{1}{768} \sum_{j=4}^{\infty} \frac{j(j-1)(j-2)(j-3)}{j!} 2^j x^j \tag{7} \end{align}$$ Now substitute $(5)$ and $(7)$ into equation $(3)$, change $i$ and $j$ to $n$, and we have the desired result $(1)$.
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