Sorry if this is a too basic question, but I didn't find an answer anywhere:

The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\infty}(T^{\ast}M^{\otimes_{s}2})$ given by $$\Delta:=\mathrm{tr}(\nabla^{2}T)$$ Now, I am confused about the coordinate expression: Many references, for example this nice lecture notes by L. Nicolaescu, state that $$\Delta:=g^{ij}(\nabla_{i}\nabla_{j}+\Gamma_{ij}^{k}\nabla_{k})$$ where $\nabla$ denotes the Levi-Civita connection. However, I am a bit puzzled by the second term. For example, let say I take the case $k=0$. Then, the operator $\Delta$ should reduce to the Laplace-Beltrami operator, which is given by $$\Delta_{\mathrm{LB}}f=g^{ij}\nabla_{i}\nabla_{j}f=g^{ij}(\partial_{i}\nabla_{j}f-\Gamma_{ij}^{k}\nabla_{j}f)=g^{ij}(\partial_{i}\partial_{j}f\underline{-\Gamma_{ij}^{k}\partial_{k}f})$$ where I used that $\nabla_{i}f=\partial_{i}f$ on zero-tensors and $\nabla_{i}T_{j}=\partial_{i}T_{j}-\Gamma_{ij}^{k}T_{k}$ on $1$-tensors. However, equation above gives $$\Delta f=g^{ij}\partial_{i}\partial_{j}f$$ i.e. the underlined part is missing. In general, I would expect that the connection Laplacian in coordinates is simply $$(\Delta T)_{i_{1}\dots i_{k}} = g^{ij}\nabla_{i}\nabla_{j}T_{i_{1}\dots i_{k}} = g^{ij}\bigg(\partial_{i}T_{i_{1}\dots i_{k}}-\sum_{l}\Gamma_{ii_{j}}^{l}\nabla_{j}T_{i_{1}\dots l\dots i_{k}}\bigg)$$ where each term $\nabla_{j}T_{i_{1}\dots l\dots i_{k}}$ has to be replaced by its usual expression for covariant derivatives of tensors. So my question can be summarized:

Why is there this additional piece $\Gamma_{ij}^{k}\nabla_{k}$ in the formula above?

I assume it is just a missunderstanding of notation or terminology. Any help appreciated.