Problem
How many ways dividing $n$ balls into $3$ buckets with the following limitations(?):
- 1st bucket contains odd number of balls.
- 2nd bucket contains a multiplication of 4 number of balls.
- 3rd bucket contains either 0 or 2 balls exactly.
I'm trying to solve this problem using Generating Functions.
Solution
Lets find the generating functions using the above limitaions: $(x+x^3+...)(1+x^4+x^8+...)(1+x^2) = x(1+x^2 +...)(1+x^4 + ...)(1+x^2)$ $= x (1+x^2) \frac{1}{1-x^2} \frac{1}{1-x^4}$
Now is the point I get stuck. What should I do next? Should I find the coefficient of $x^n$? If so, how?