 Generating function of the Catalan numbers

The ordinary generating function $C(x)={\sum }_{n\ge 0}{C}_n{x}^n$ packages the Catalan recurrence. Its Cauchy product gives the formal power series identityThe constant term is $C_0=1$. Solving the quadratic equation chooseswith the formal power series square root having constant term one. The apparent quotient at $x=0$ has constant term one after cancellation; the other quadratic solution has a pole and cannot be this ordinary generating function. The equation records the empty object or an outer unit with two ordered smaller objects.