Coefficients of series given by generating function How to find the coefficients of this infinite series given by generating function.$$g(x)=\sum_{n=0}^{\infty}a_nx^n=\frac{1-11x}{1-(3x^2+10x)}$$
I try to expand like Fibonacci sequences using geometric series and binomial theorem but without any success.  

Use (1) the fact that $1-3x^2-10x=(1-ax)(1-bx)$ for some $a$ and $b$, (2) the fact that
$$
\frac{1-11x}{(1-ax)(1-bx)}=\frac{c}{1-ax}+\frac{d}{1-bx},
$$
for some $c$ and $d$, and (3) the fact that, for every $e$,
$$
\frac1{1-ex}=\sum_{n\geqslant0}e^nx^n.
$$
Then put all these pieces together to deduce that, for every $n\geqslant0$,
$$
a_n=c\cdot a^n+d\cdot b^n.
$$
Edit: You might want to note that $g(0)=1$ hence $a_0=1$, which yields $c+d=1$. 
Likewise, $1/(1-u)=1+u+o(u)$ when $u\to0$ hence $g(x)=(1-11x)(1+10x+o(x))=1-x+o(x)$. This shows that $g'(0)=-1$ thus $a_1=-1$ and $ca+(1-c)b=-1$, that is, $c=(1+b)/(b-a)$. Finally, $a$ and $b$ are the roots of the denominator $1-3x^2-10x$ and, for every $n\geqslant0$,
$$
a_n=\frac{(1+b)\cdot a^n-(1+a)\cdot b^n}{b-a}.
$$
