In fact, the only non-trivial Christoffel symbols at $p$ are

(3.320)

$$\Gamma_{ij}^{1}(0)=\partial_{i}g_{j\bar{1}}(0),\ \ \Gamma_{\bar{i}\bar{j}}^{\bar{1}}=\partial_{\bar{i}}g_{1\bar{j}}(0)$$

for $i,j\geq 2$. This is due to the constraint that the equation $w_{1}=0$ defines $H$, which prevents us from using substitutions like

(3.321)

$$w_{1}=z_{1}+\sum\limits_{i,j=2}^{n-1}C_{1ij}z_{i}z_{j}.$$

Intrinsically, $\{\Gamma^{1}_{ij}\}_{i,j\geq 2}$ captures the second fundamental form of the complex hypersurface $H$ at $p$.