Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 \qquad \text{where} \qquad f_{ij} := \frac{\partial^{i+j} f(x,y)}{\partial x^i y^j}|_{(0,0)}.$$ What this means is that at the origin, the derivative of $f$ vanishes, plus the tangent vector $ \partial_x $ is in the Kernel of the Hessian of $f$. Moreover, the Hessian is not identically zero.

Hence, we can write the Taylor expansion of $f$ as $$ f = A_0(x) + A_1(x) y + A_2(x) y^2 + \ldots, \qquad \text{where} \qquad A_2(0) \neq 0.$$ Now it is easy to see (using the fact that $A_2(0) \neq 0$ and the implicit function theorem) that one can make a change of coordinates $y = \tilde{y}+ B(x)$, so that $f$ becomes $$f = \hat{A}_0(x) + \hat{A}_2(x) \tilde{y}^2 + \ldots \qquad \text{ie} \qquad \hat{A}_1(x) =0.$$ This is an explicit procedure, ie $B(x)$ and $\hat{A}_0(x)$ can be computed as power series in $x$ and the coefficients of $x^n$ will just be some functions of the original $f_{ij}$. Suppose $$ \hat{A}_0(x) = C_k x^k + C_{k+1} x^{k+1} + \ldots $$

My question: Is the quantity $C_k$ is invariant under $y \rightarrow y+x$?

Note that under the transformation $y \rightarrow y+x$, the $f_{ij}$ are all going to change. But the precise combination that makes up $C_k$, is going to be unchanged. That is the claim.

One can explicitly check this claim for each $k$. I believe there should be some obvious reason why this is true from the way those $C_k$ were obtained. But I don't see it immediately.

I intuitively expect this to be true because of the following reason: The direction $\partial_x$ is a special direction, ie the hessian vanishes along $\partial_x$. But, there is nothing really special about the direction $\partial_y$. It belongs to the quotient space $ T\mathbb{C}^2/ < \partial_x>$ (and not the orthogonal complement with respect to some metric).

$\textbf{Not sure if relevant, but might help with the understanding:}$
The significance of these $C_k$ is that it gives us a necessary and sufficient criteria for the curve $f=0$ to have a singularity of type $A_{k-1}$ at the origin (ie it can be written as $v^2 + u^{k} =0$ after a change of coordinates). If $C_i =0$ till $i=k-1$ and $C_{k} \neq 0$ then, the curve has an $A_{k-1}$ singularity at the origin as can easily be seen by a change of coordinates.