Call the generating function \(A\text{.}\) Compute \(A - xA = 4 + x + 2x^2 + 3x^3 + 4x^4 + \cdots\text{.}\) Thus \(A - xA = 4 + \dfrac{x}{(1-x)^2}\text{.}\) Solving for \(A\) gives \(\d\frac{4}{1-x} + \frac{x}{(1-x)^3}\text{.}\)