Determine the generating function A(x)=∑n=0∞anxnA(x) = \sum_{n=0}^{\infty} a_n x^nA(x)=∑n=0∞anxn for an=n⋅2na_n = n \cdot 2^nan=n⋅2n.
2x(1−2x)2\frac{2x}{(1-2x)^2}(1−2x)22x
x(1−2x)2\frac{x}{(1-2x)^2}(1−2x)2x
2x1−4x+4x2\frac{2x}{1-4x+4x^2}1−4x+4x22x
1(1−2x)2\frac{1}{(1-2x)^2}(1−2x)21