= Solution
For $n\geq1$, let $p_L,p_R$ be the <interpolation polynomials> of degree at most $n-1$ on nodes $x_0,\ldots,x_{n-1}$ and $x_1,\ldots,x_n$, respectively. The polynomial
$$
p(x)=\frac{(x-x_0)p_R(x)-(x-x_n)p_L(x)}{x_n-x_0}
$$
has degree at most $n$. At $x_0$ it equals $p_L(x_0)$; at $x_n$ it equals $p_R(x_n)$. At each common node both interpolants equal $f$, and the displayed weights sum to one, so $p$ also equals $f$ there. Thus it is the full interpolant by uniqueness.
Its coefficient of $x^n$ is the difference of the leading coefficients of $p_R,p_L$ divided by $x_n-x_0$. By the definition of a <divided difference>,
$$
\boxed{f[x_0,\ldots,x_n]=\frac{f[x_1,\ldots,x_n]-f[x_0,\ldots,x_{n-1}]}{x_n-x_0}}.
$$
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