Determine the generating function A(x)=∑n=0∞anxnA(x) = \sum_{n=0}^{\infty} a_n x^nA(x)=∑n=0∞anxn for the recurrence an=2an−1+na_n = 2a_{n-1} + nan=2an−1+n with a0=1a_0 = 1a0=1.
A(x)=1−x+x2(1−x)2(1−2x)A(x) = \frac{1-x+x^2}{(1-x)^2(1-2x)}A(x)=(1−x)2(1−2x)1−x+x2
A(x)=1−x(1−2x)2A(x) = \frac{1-x}{(1-2x)^2}A(x)=(1−2x)21−x
A(x)=1(1−x)(1−2x)A(x) = \frac{1}{(1-x)(1-2x)}A(x)=(1−x)(1−2x)1
A(x)=1−x+x2(1−x)(1−2x)2A(x) = \frac{1-x+x^2}{(1-x)(1-2x)^2}A(x)=(1−x)(1−2x)21−x+x2