I have $$A(x) = \sum\_{n=0}^\infty a\_n x^n$$ and $$A(x) = (1+x)/(1+7x+6x^2)$$

I need to find $a\_0,a\_1,a\_2,a\_3$.

I multiply on each side and I get

$$
(1+7x+6x^2) (a\_0+a\_1x+a\_2x^2+a\_3x^3+\ldots)=1+x
$$

Now I get
$$
a\_0+a\_1x+7a\_0x+\ldots = a\_0 + x(a\_1+7a\_0) = 1+x
$$

and I can compare the two sides which gives me
$$
a\_0 = 1, \qquad a\_1=1-7a\_0=-6, \qquad a\_2=0, \qquad a\_3=0.
$$

This makes sense to me but according to my homework the correct result should be $a\_0 = 1$, $a\_1 = -6$, $a\_2 = 36$ and $a\_3 = -216$.

What am I doing wrong?