Find the number of possibilities to divide $n$ balls into $3$ cells, such that:
- In the first cell there must be at least one ball.
- No limitations for middle cell.
- In right cell, the number of balls isn't dividable by $3$.
$$f(x) = \frac{x}{{1 - x}} \cdot \frac{1}{{1 - x}} \cdot \left( ? \right)$$
The first two are relatively easy. How to convert the third demand into a generating function?
More specifically, What is the power series of:
$$y = (x + {x^2} + {x^4} + {x^5} + ...)$$
Thanks.
Hint : Subtract the sequence $x^3+x^6+x^9+...$ from $x+x^2+x^3+...$
Both sequences are geometric.