Is the Riemann curvature tensor the only tensor that can be constructed from the metric tensor and its first and second derivatives?

Question

I am reading Gravitation and Cosmology by Steven Weinberg. On page 133, he says

$R^{\lambda}_{\phantom{x}\mu\nu\kappa}$ is the only tensor that can be constructed from the metric tensor and its first and second derivatives, and is linear in the second derivatives.

I have never heard this. I'd like to read more about this property.

Can someone either (1) explain why this seems like a natural consequence given what we know about the Riemann tensor or (2) direct me to a resource so I can read about it myself?