= Generating function of the Catalan numbers
{title2=$C(x)=\frac{1-\sqrt{1-4x}}{2x}$}
The <ordinary generating function> $C(x)=\sum_{n\geq0}C_nx^n$ packages the <Catalan recurrence>. Its <Cauchy product> gives the <formal power series> identity
$$
C(x)=1+xC(x)^2.
$$
The constant term is $C_0=1$. Solving the <quadratic equation> chooses
$$
C(x)=\frac{1-\sqrt{1-4x}}{2x},
$$
with 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.
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