Taylor series for exponential map on "famous" manifolds (as the sphere)

Question

I read many papers and books that give a Taylor series for the exponential map on a smooth manifold $(\mathcal{M},\nabla)$ e.g. https://arxiv.org/abs/1205.2868v1. It can be possible to find somewhere explicit formulas of that series for manifold as the 2D sphere, also up to the $2^{nd}$ order? Thank you!