A simple expansion Where I am doing wrong? if any one can tell!!!
I have done this equation as
$$y ={x\over 1+(x-x^4+x^7)^3} = x [1+(x-x^4+x^7)^3]^{-1}$$
$$y= x[ 1-(x-x^4+x^7)^3+...]$$
$$y= x[ 1-(x^3+x^{12}+x^{21}-3x^6+3x^{9}+3x^9+3x^{15}+3x^{15}-3x^{18})+...] $$ 
$$y= x[ 1-x^3-x^{12}-x^{21}+3x^6-3x^9-3x^9-3x^{15}-3x^{15}+3x^{18}+...]$$
$$y= x-x^4+3x^7+...$$
But in book it is written as,
$$y= x-x^4+4x^7+...$$

This may be a typo in the book. If you expand the reciprocal one term further there is a term $x^7.$ Perhaps the author mistakenly wrote your $3x^6$ as $3x^7$ and then grouped it with the extra $x^7.$
Do you want to share the book's reference?
