This is my first time on the Mathematics Stack Exchange. I am currently studying for my IIT entrance exam, and while studying and browsing the web, I came across an interesting problem.
Let $a,b,c,d \ge 0$. Find a closed form for the expression: $$\sum_{a+b+c+d = n} 2^{a+2b+3c+4d}$$ in terms of $n$.
At the bottom, it said that a possible hint might be that I should use generating functions. I've been working for the past couple of hours and can't seem to figure out what the generating function of this expression would be.
Any help would be greatly appreciated!
Thank you, Rohit
Hint: If you multiply out $$\dfrac{1}{1-2x} \cdot \dfrac{1}{1-2^2x} = \left(\sum_{a = 0}^{\infty}2^ax^a\right) \cdot \left(\sum_{b = 0}^{\infty}2^{2b}x^b\right),$$ the $x^n$ coefficient is the sum of $2^a \cdot 2^{2b} = 2^{a+2b}$ over all integers $a,b \ge 0$ such that $a+b = n$.
Can you extend this idea to your problem?