determine the sequence generated by (6-2x)/(1+2x-15x^2)
Use partial fractions decomposition. It should give you that
$\displaystyle \frac{6-3x}{-(3x-1)(5x+1)}=\frac{33}{8(5x+1)}-\frac{15}{8(3x-1)}$
Now use that $\displaystyle \frac{1}{1-z}=\sum_{n=0}^{\infty}z^n$
IF this is indeed what you are looking for the first term should give that $\displaystyle \sigma_n=\frac{33}{8}(-1)^n5^n$. Now find $\displaystyle \delta_n$ for the second one. And your function will be the generator of $\displaystyle \sigma_n+\delta_n$