<s>[INST] How to do power series expansion What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$?

How do I do power series expansions?
 [/INST] suppose $$\frac{1}{1-x-x^4} = 1 + x + a_2x^2 + a_3x^3 + \cdots.  $$ then we have $$1 =(1-x-x^4)(1 + x + a_2x^2 + a_3x^3 +\cdots)=1+(a_2-1)x^2 + (a_3-a_2)x^3+\\(a_4-a_3-1)x^4+\cdots+(a_n-a_{n-1} -a_{n-4})x^n+\cdots  $$  equating the coefficients of $x^2, x^3, \cdots$, we find that 
$$ a_0 = 1, a_1 = 1, a_2 = 1, a_3 =1, a_4=2 \text{ and } a_n=a_{n-1} +a_{n-4} $$
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline
n&0&1&2&3&4&5&6&7&8&9&10&11&12 \\ \hline
a_n&1&1&1&1&2&3&4&5&7&10&14&19&26  \\ \hline
\end{array}$
$$ \text{ i found }19 \text{ to be the coefficient of } x^{11}.$$
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