So there are three shareholders and 2n shares to split among them

let be the number of ways the shares can be split such that no single person controls half the total or more

The question asks me to find the generating function A(x) for the sequence

I know but I need to find the generating function not the actual closed formula

I've reverse engineered it to be

But I need to find it coming from the other direction haha

Could anyone explain how to do this?

thanks

JB