# Sufficient conditions for the coefficients of a generating function to dominate those of its square

Let $$f(z)$$ be a generating function (so in particular, its power series coefficients are nonnegative). I am interested in conditions which would ensure that for every $$n$$, the coefficient of $$z^n$$ in $$f$$ is greater than or equal to the coefficient of $$z^n$$ in $$f^2$$. Equivalently, I would like the generating function $$f-f^2$$ to have nonnegative coefficients.

One necessary condition is that $$f$$ have no constant term, so $$f(0)=0$$. Another necessary condition is that $$f$$ have infinitely many terms; polynomials will never satisfy this condition. But these necessary conditions are very far from sufficient.

For an example of a generating function that does satisfy this condition, let $$f$$ denote the generating function for the shifted Catalan numbers, $$f=\frac{1-\sqrt{1-4z}}{2}=z+z^2+2z^3+5z^4+14z^5+\cdots.$$ Then we have that $$f-f^2=z$$, which satisfies the condition.

However the condition seems to be quite fragile. If we instead take $$f=\frac{1-2z-\sqrt{1-4z}}{2z}=z+2z^2+5z^3+14z^4+42z^5+\cdots$$ then we have $$f-f^2=z+z^2+z^3-6z^5-\cdots$$, which does not satisfy it.

The only other necessary condition I know of is that the dominant singularity of $$f$$ cannot be a pole, because if it were a pole then (by Pringsheim's Theorem) there would be a real number $$z_0>0$$ such that $$f(z_0)>1$$, and thus we would have $$f^2(z_0)>f(z_0)$$, so the condition could not hold.

Other than a few specific examples (e.g., the shifted Catalan numbers above), I know of no sufficient conditions.

Take any series $$g$$ with zero free term and all other coefficients being nonnegative, and solve $$f-f^2=g$$ with respect to $$f$$, i.e., $$f=\frac{1-\sqrt{1-4g}}2.$$ This is the general form of all such $$f$$, and it can be easily seen that the coefficients of $$f$$ are nonnegative as well.