I'm just learning about Generating Functions so my question might not completely make sense (in that case, I apologize). I want to know whether there exist at least one number that is generated by both generating functions $f:=\cfrac{1}{\sqrt{1-4x}}$ and $g:=\cfrac{x^k}{(1-x)^{k+1}}$ where $k=3,4,...,n$. If there exist such number(s) (I don't know how to determine that) what are they?
Update
- By "a number that is generated by both generating functions" I meant that a number exists that is the coefficient of both generating functions.
- There exists solutions (two equal coefficients) if k=3,4,5 ... However how can I show that there exist only 1 solution if k=n? Or there exist multiple solutions? Or there exist no solutions?
The Taylor series of a generating function gives you the coefficients (I guess that's what you mean by "number generated by the generating function) of a generating function. If you're too lazy to work out the expansions by hand, then you can let Wolfram Alpha do the work. After a little of try-and-error you should be able to find coefficients that are in both expansions.