Partial fractions with binomials

Partial fractions. $A(1-2x)+B(1-x)=1$, so $A+B=1$ and $-2A-B=0$. Solving: $A=-1$, $B=2$.

Series. $\frac{1}{(1-x)(1-2x)}=-\frac{1}{1-x}+\frac{2}{1-2x}=-\sum x^n+2\sum 2^nx^n=\sum_{n\ge0}(2^{n+1}-1)x^n$.

Coefficient. $a_n=2^{n+1}-1$.

Conclusion. $\frac{1}{(1-x)(1-2x)}=\sum_{n\ge0}(2^{n+1}-1)x^n$.

Alternative. Using binomial: $a_n=\sum_{k=0}^{n}2^k=2^{n+1}-1$.