= Lagrange inversion theorem
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= Lagrange inversion
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{synonym}
Over a <field> of <characteristic> zero, let $w(x)$ be the unique <formal power series> with constant coefficient zero satisfying $w=x\phi(w)$, where $\phi(0)\ne0$. <Lagrange inversion> gives, for $n\ge1$,
$$
[x^n]F(w(x))=\frac1n[t^{n-1}]F'(t)\phi(t)^n.
$$
Here the brackets denote <coefficient extraction>. The theorem obtains coefficients of an implicitly defined series without solving for it explicitly.
For $w=x(1+w)^2$ and $F(t)=t$, it gives $[x^n]w=\binom{2n}{n-1}/n=\binom{2n}{n}/(n+1)$, the <Catalan numbers> for $n\ge1$. This is useful when a recursive counting construction naturally gives an implicit <generating function>.
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