When writing the equation of the geodesics of a Riemannian manifold in local coordinates I have (I consider the Einstein's summation convention):
$$\frac{d^2x^\mu}{dt^2} + \Gamma^\mu_{\nu \rho} \frac{dx^\nu}{dt} \frac{dx^\rho}{dt} = 0$$
If a consider a symmetric connection as the Levi-Civita connection I have that $\Gamma^\mu_{\rho \nu} = \Gamma^\mu_{\nu \rho}$. Does it mean that I have to sum twice the Cristoffel symbols (one for $\nu \rho$ and another one for $\rho \nu$)?
Yes, the summation here is over the $n^2$-dimensional array $1 \leq \nu \leq n, 1 \leq \rho \leq n$. You can also write the second term as
$$ \sum_{\nu = 1}^n \Gamma^{\mu}_{\nu \nu} \frac{dx^{\nu}}{dt} \frac{dx^{\nu}}{dt} + 2 \sum_{1 \leq \nu < \rho \leq n} \Gamma^{\mu}_{\nu \rho}\frac{dx^{\nu}}{dt} \frac{dx^{\rho}}{dt}. $$