What's the Taylor series for $f(x+h,y+k)$? My course notes give the Taylor series up to degree 2 as
$$f(x+h,y+k)\approx f(x,y)+\frac{\partial f}{\partial x}h+\frac{\partial f}{\partial y}k+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}h^2+\frac{\partial^2 f}{\partial x \partial y}hk+\frac{1}{2}\frac{\partial^2 f}{\partial y^2}k^2.$$
It looks like the formula could be something like
$$f(x+h,y+k)=\sum_{i=0}^\infty \left(\sum_{j=0}^i \left(\frac{\partial f^i}{\partial (x^{i-j}y^j)}\cdot\frac{h^{i-j}k^j}{i!} \right)\right),$$
except, in degree 2, the factorial doesn't seem to apply equally to the mixed term, and I can't see what the pattern would be without more terms.

If we denote $x^1 := x, x^2 := y$ and $h^1 := h, h^2 := k$, then the terms of total degree $i$ are those of the sum
$$\frac{1}{i!} \sum_{k_i = 1}^2 \cdots \sum_{k_1 = 1}^2 \frac{\partial^i f}{\partial x^{k_1} \cdots \partial x^{k_i}}(x^1_0, x^2_0) h^{k_1} \cdots h^{k_i} .$$
Expanding this in the case $i = 2$ indeed gives the quadratic terms from the formula in your notes.
If $f$ has continuous $k$th derivatives, then we may reorganize this sum by grouping terms according to the number of indices $k_j$ that are $1$ and the number that are $2$. For any $j \in \{0, \ldots, i\}$ among the $2^i$ strings $(h^{k_1}, \ldots, h^{k_i})$ there are $i \choose j$ with $i - j$ of the $k_1$ equal to $1$ and $j$ equal to $2$. Thus, indexing the sum by $j$ gives
$$\frac{1}{i!} \sum_{j=0}^i {i \choose j} \frac{\partial^i f}{\partial (x^1)^{i - j}  \,\partial (x^2)^j}(x^1_0, x^2_0) (h^1)^{i - j} (h^2)^j .$$
Rewriting using our original variable names gives the easier-to-read expression
$$\frac{1}{i!} \sum_{j = 0}^i {i \choose j} \frac{\partial^i f}{\partial x^{i - j}  \,\partial y^j}(x_0, y_0) h^{i - j} k^j ,$$
and so the multivariable Taylor series is
$$\sum_{i = 0}^\infty \frac{1}{i!} \sum_{j = 0}^i {i \choose j} \frac{\partial^i f}{\partial x^{i - j}  \,\partial y^j}(x_0, y_0) h^{i - j} k^j .$$
One can make this proof and the formulae a little cleaner using multi-index notation, in which case the final formula is
$$\sum_{i = 0}^\infty \sum_{|\alpha| = 1}^i \frac{1}{|\alpha|!} \frac{\partial^\alpha f}{\partial x^\alpha}(x) \,h^\alpha$$
