consider

𝑦=𝑛=0𝑎𝑛𝑥𝑛

then

𝑦=𝑛=1𝑛𝑎𝑛𝑥𝑛1=𝑛=0(𝑛+1)𝑎𝑛+1𝑥𝑛

and

𝑦=𝑛=2𝑛(𝑛1)𝑎𝑛𝑥𝑛2=𝑛=0(𝑛+2)(𝑛+1)𝑎𝑛+2𝑥𝑛

then solve the equation by normal calculation