find the taylor series of a function of two variables

Hello,

I was given this practice question...

Find the Taylor Series for f(x,y)=1/(2+xy²).

I though this would be easy but I can't figure out how they got the answer that they did in the back of the book.

I tried to find the pattern by finding f1 and f2, then f11, f12 and f22. Then I put it into the form of a taylor polynomial to find the pattern.

After this point I ended up with 1/2 + (3xy²)/4 + (5x²y⁴ - 4xy²)/8 ...

The answer in the back of the book is: Sum n=0 ->inf (-1)ⁿ xⁿy²ⁿ/2⁽ⁿ⁺¹⁾

I understand the (-1)ⁿ and the 2⁽ⁿ⁺¹⁾ and for the most part I can see the xⁿy²ⁿ part also but something is off somewhere.

If anyone can give me any feed back I would really appreciate it.